Home Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
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Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks

  • Li Feng and Jing Wang EMAIL logo
Published/Copyright: November 26, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, we study the inviscid limit of the solution to the Cauchy problem of a one-dimensional viscous conservation law, where the second-order term is nonlinear. Under the assumption that the inviscid equation admits a piecewise smooth solution with two noninteracting entropy shocks, we prove that the solution of the viscous equation converges uniformly to the piecewise smooth inviscid solution away from the shocks, even the strength of shocks is not small.

Keywords: shock layer; viscous shocks; matched asymptotic expansion; nonlinear stability; energy estimates
MSC 2010: 35L50; 35L60; 35L65; 35K59; 35K65

1 Introduction

In fluid mechanics, it is well known that a fluid is composed of a large number of molecules that continuously make thermal motion and have no fixed equilibriums. When there is a relative sliding among adjacent layers of a fluid, shear stress, which is known as the viscous stress, will occur due to the interaction of these molecules. Actual fluids in nature are all viscous fluids. The motion of one-dimensional (1D) viscous fluids is described by the viscous conservation laws:

(1.1) ∂ t u ε + ∂ x f ( u ε ) = ε ∂ x ( B ( u ε ) ∂ x u ε ) , u ε ∈ R n , x ∈ R 1 , t > 0 ,

where u ε denotes the density, velocity, or other physical quantities, and B is called the viscosity matrix. In the case where the viscosity is small and the relative sliding velocity is not large, the viscous stress will be small, and the thermal conduction and diffusion effects can be neglected; thus, the fluid is considered to be an ideal fluid. In this way, ideal models are only approximations of real fluids. The motion of ideal fluids can be represented in the following system of conservation laws:

(1.2) ∂ t u + ∂ x f ( u ) = 0 , u ∈ R n , x ∈ R 1 , t > 0 .

In many physical phenomena and their numerical computations, to study the asymptotic relation between the viscous parabolic system (1.1) and its associated inviscid hyperbolic equations (1.2) in the limit of small dissipations for various viscous terms is of considerable significance. In general, the motion of the fluid is always in a domain with boundaries, and solutions of conservation laws will produce singularities except for some special cases (see [1–4], such as shock waves). The viscous flow will display singular behavior in the limit of small viscosity [5]. So the topic of vanishing viscosity limit has attracted much attention especially for problems in the presence of shocks and boundaries (see [6–14] and references therein).

The structure of the viscosity matrix also plays an important role in the proof of vanishing viscosity limits. The case of identity viscosity matrix has been studied in [7–9,12,14,15] for both Cauchy problems and initial boundary value problems. Among these studies, the vanishing viscosity limit of solutions to a 1D quasilinear parabolic system in the presence of a single shock discontinuity is studied in [7], where the solution of the inviscid problem is piecewise smooth with a single shock satisfying the entropy condition. It is proved that if the strength of the single shock is small for the underlying inviscid flow, then the solution to the viscous system will converge to the solution of the inviscid system away from the shock as the viscosity coefficient ε tends to zero. When the viscosity term is nonlinear, the boundary-layer problems have been discussed in [10,11,13,16–19], where the nonlinear viscosity led to more complicated analyses and computations.

In this article, enlightened by [7], we consider the Cauchy problem of the scalar case of (1.2). Precisely speaking, we study the asymptotic equivalence between the viscous problem with the general viscous term

(1.3) ∂ t u ε + ∂ x f ( u ε ( x , t ) ) = ε ∂ x ( b ( u ε ) ∂ x u ε ) , t > 0 , x ∈ R 1 , u ∈ R 1 ,

(1.4) u ε ( x , t = 0 ) = u 0 ε ( x ) ,

and the inviscid problem

(1.5) ∂ t u + ∂ x f ( u ( x , t ) ) = 0 , t > 0 , x ∈ R 1 ,

(1.6) u ( x , t = 0 ) = u 0 0 ( x ) ,

where f is smooth, and we require that

(1.7) f ′ ( u ) < 0 and f ″ ( u ) > 0 .

Moreover, b ( u ) in the viscous term is a nonnegative smooth function, and there exists g ( u ( x , t ) ) satisfying

(1.8) g ′ ( u ( x , t ) ) = b ( u ( x , t ) ) .

We assume that (1.5)–(1.6) admits a piecewise smooth solution, which satisfies the following conditions:

  1. u ( x , t ) is a distributional solution of the hyperbolic equation (1.5) in R × [ 0 , T ] with T > 0 ;

  2. there are two disjoint smooth shock curves x = s i ( t ) , i = 1 , 2 , 0 ≤ t ≤ T , so that u ( x , t ) is sufficiently smooth at any point x ≠ s i ( t ) , i = 1 , 2 ;

  3. the limits

    (1.9) ∂ x k u ( s i ( t ) − 0 , t ) = lim x → s i ( t ) − ∂ x k ( u ( x , t ) ) , and

    (1.10) ∂ x k u ( s i ( t ) + 0 , t ) = lim x → s i ( t ) + ∂ x k ( u ( x , t ) ) ,

    exist and are finite for t ≤ T and i = 1 , 2 ;

  4. the Lax entropy condition is satisfied at x = s i ( t ) , i = 1 , 2 , i.e.,

    (1.11) f ′ ( u ( s i ( t ) − 0 , t ) ) > d s i d t > f ′ ( u ( s i ( t ) + 0 , t ) ) .

    Then, u ( x , t ) is a unique piecewise smooth solution with two noninteracting shocks of (1.5) for [ 0 , T ] .

For m ( x ) ∈ C 0 ∞ ( R ) satisfying 0 ≤ m ( x ) ≤ 1 ,

(1.12) m ( x ) = 1 , ∣ x ∣ ≤ 1 , 0 , ∣ x ∣ ≥ 2 , h ( x ) , 1 < ∣ x ∣ < 2 ,

with 0 < h ( x ) < 1 being a smooth function. Let m i = m ( x − s i ( t ) ε γ ) , i = 1 , 2 , γ ∈ ( 6 7 , 1 ) . In [7], the initial data of the two types of equations are not involved. Here, to avoid the phenomenon of the initial layer, we assume that the initial data u 0 ε ( x ) have the following asymptotic expansion:

(1.13) ‖ u 0 ε − m 1 ∑ i = 0 3 ε i u s i ( ξ , 0 ) + m 2 ∑ i = 0 3 ε i u ¯ s i ( η , 0 ) + ( 1 − m 1 − m 2 ) ∑ i = 0 3 ε i u 0 i ( x ) ‖ L 1 ( R ) ∩ H 2 ( R ) ≤ C ε 2 ,

where u s i ( ξ , 0 ) , u ¯ s i ( η , 0 ) , and u 0 i are the known functions in the process of expansions near or away from the shocks. In this way, the initial data u 0 ε is a more general version than the so-called “well prepared” initial data in the previous research.

We study the evolution and structure of viscous shock layers, which are related to the viscous shock profiles [20], and their interaction with interior hyperbolic inviscid flow, and show that the uniform convergence of the viscous solutions to the piecewise smooth inviscid flow away from the shock discontinuities without the assumption that the strength of the shocks is small. We obtain the results of the inviscid limit for the Cauchy problem as follows:

Theorem 1.1

Suppose that the inviscid equation (1.5) is strictly hyperbolic, and there exists a constant ε 0 > 0 such that u ( x , t ) is a piecewise smooth solution with two noninteracting shocks of (1.5)–(1.6) up to time T > 0 , then for each 0 ≤ ε ≤ ε 0 , the nonlinear viscous equations (1.3)–(1.4) with the initial data satisfying (1.13) have a unique smooth solution u ε ( x , t ) ∈ C 1 ( [ 0 , T ] ; H 2 ( R ) ) such that

(1.14) sup 0 ≤ t ≤ T ∫ R ∣ u ε ( ⋅ , t ) − u ( ⋅ , t ) ∣ 2 d x ≤ C ε σ ,

(1.15) sup 0 ≤ t ≤ T ∣ x − s i ( t ) ∣ ≥ ε γ ∣ u ε ( ⋅ , t ) − u ( ⋅ , t ) ∣ ≤ C ε , i = 1 , 2 ,

where 3 4 ≤ σ ≤ 1 , 6 7 ≤ γ ≤ 1 , and C is a positive constant.

By the method of multiple-scale asymptotic expansions, we first construct the four-term approximate solutions of the viscous equations (1.3) in different regions, and then match them up to obtain the approximate solution in the whole space. For the leading order functions with fast variables, we derive the explicit solution formulas of the shock profiles from the ordinary differential equations, so that we can clearly obtain the exponential decay property. Then, similar to [14], we decompose the viscous solution into two parts with the term ε 1 ⁄ 2 + δ φ , with 0 < δ < 1 ⁄ 2 , to carry out energy estimates of the error equation of φ , and we prove that the approximate solution converges uniformly to the viscous solution u ε ( x , t ) . Consequently, the vanishing viscosity limit can be verified away from the shocks. Moreover, the method here can also be applied to the case of finite noninteracting shocks.

2 Construction of the approximate solution

In this section, we use the method of matched asymptotic expansions to give a detailed construction of the approximate solution u ¯ a ( x , t ) to (1.3)–(1.4).

2.1 Outer expansion

Away from the shocks, we expand the viscous solution as

(2.1) u ε ∼ u 0 ( x , t ) + ε u 1 ( x , t ) + ε 2 u 2 ( x , t ) + ε 3 u 3 ( x , t ) + … .

Substituting it into (1.3) and equating the coefficients of different orders of ε give

(2.2) O ( 1 ) : ∂ t u 0 + ∂ x f ( u 0 ) = 0 ,

(2.3) O ( ε ) : ∂ t u 1 + ∂ x ( f ′ ( u 0 ) u 1 ) = ∂ x ( b ( u 0 ) ∂ x u 0 ) ,

(2.4) O ( ε 2 ) : ∂ t u 2 + ∂ x ( f ′ ( u 0 ) u 2 ) = ∂ x 2 ( b ( u 0 ) u 1 ) − 1 2 ∂ x ( f ″ ( u 0 ) ( u 1 ) 2 ) ,

(2.5) O ( ε 3 ) : ∂ t u 3 + ∂ x ( f ′ ( u 0 ) u 3 ) = ∂ x 2 ( b ( u 0 ) u 2 ) + 1 2 ∂ x 2 ( b ′ ( u 0 ) ( u 1 ) 2 ) − ∂ x ( f ″ ( u 0 ) ( u 1 ⋅ u 2 ) ) − 1 6 ∂ x ( f ‴ ( u 0 ) ( u 1 ) 3 ) .

We now require the initial data of the outer functions u i ( i = 0 , … , 3 ) , which are as follows:

(2.6) u 0 ( x , 0 ) = u 0 0 ( x ) ,

(2.7) u i ( x , 0 ) = u 0 i ( x ) , i = 1 , 2 , 3 .

Note that problems (2.2)–(2.6) for the leading order outer function are exactly the nonlinear problems (1.5)–(1.6). Therefore, we take u 0 to be the given unique piecewise smooth solution with two noninteracting shocks of (1.5)–(1.6). We assume the initial data u 0 i ( x ) , i = 1 , 2 , 3 are smooth, then the functions u 1 , u 2 , and u 3 determined by the linear equations are generally discontinuous at shock curves x = s i ( t ) , i = 1 , 2 , but smooth up to the shocks [2,4].

2.2 Expansions near shocks

Near the shock x = s 1 ( t ) , we approximate u ε as

u ε ( x , t ) ∼ u s 0 ( ξ , t ) + ε u s 1 ( ξ , t ) + ε 2 u s 2 ( ξ , t ) + ε 3 u s 3 ( ξ , t ) + … ,

where

ξ = x − s 1 ( t ) ε + δ 1 ( t , ε ) ,

and δ 1 is a disturbance of the shock position, which is to be determined. Assume that the expansion of δ 1 ( t , ε ) is

δ 1 ( t , ε ) = δ 1 0 ( t ) + ε δ 1 1 ( t ) + ε 2 δ 1 2 ( t ) + … .

Substituting it into (1.3) and equating the coefficients of different orders of ε give

O 1 ε : ∂ ξ f ( u s 0 ) − s 1 ˙ ( t ) ∂ ξ u s 0 − ∂ ξ ( b ( u s 0 ) ∂ ξ u s 0 ) = 0 , O ( 1 ) : ∂ ξ 2 ( b ( u s 0 ) u s 1 ) − ∂ ξ ( f ′ ( u s 0 ) u s 1 ) + s 1 ˙ ( t ) ∂ ξ u s 1 = δ 1 0 ˙ ∂ ξ u s 0 + ∂ t u s 0 , O ( ε ) : ∂ ξ 2 ( b ( u s 0 ) u s 2 ( ξ , t ) − ∂ ξ ( f ′ ( u s 0 ) u s 2 ) + s 1 ˙ ( t ) ∂ ξ u s 2 = δ 1 1 ˙ ∂ ξ u s 0 + δ 1 0 ˙ ∂ ξ u s 1 + ∂ t u s 1 + 1 2 ∂ η ( f ″ ( u s 0 ) ( u s 1 ) 2 ) − 1 2 ∂ η 2 ( b ′ ( u s 0 ) ( u s 1 ) 2 ) , O ( ε 2 ) : ∂ ξ 2 ( b ( u s 0 ) u s 3 ) − ∂ ξ ( f ′ ( u s 0 ) u s 3 ) + s 1 ˙ ( t ) ∂ ξ u s 3 = δ 1 2 ˙ ∂ ξ u s 0 + δ 1 1 ˙ ∂ ξ u s 1 + δ 1 0 ˙ ∂ ξ u s 2 + ∂ t u s 2 + ∂ ξ ( f ″ ( u s 0 ) ( u s 1 ⋅ u s 2 ) ) + 1 6 ∂ ξ ( f ‴ ( u s 0 ) ( u s 1 ) 3 ) − ∂ ξ [ b ′ ( u s 0 ) ∂ ξ ( u s 1 ⋅ u s 2 ) ] − 1 6 ∂ ξ 2 ( b ″ ( u s 0 ) ( u s 1 ) 3 ) ,

where s 1 ˙ ( t ) = d s 1 d t , δ 1 i ˙ = d δ 1 i d t , i = 1 , 2 , 3 .

In the matching region, we expect both the expansion near the shock and the expansion outside to be effective, so they must agree with each other in the matching region. Then, the following relations, the so-called matching conditions hold if ξ → ± ∞ :

(2.8) u s 0 ( ξ , t ) = u 0 ( s 1 ( t ) ± 0 , t ) + o ( 1 ) ,

(2.9) u s 1 ( ξ , t ) = u 1 ( s 1 ( t ) ± 0 , t ) + ( ξ − δ 1 0 ) ∂ x u 0 ( s 1 ( t ) ± 0 , t ) + o ( 1 ) ,

(2.10) u s 2 ( ξ , t ) = u 2 ( s 1 ( t ) ± 0 , t ) + ( ξ − δ 1 0 ) ∂ x u 1 ( s 1 ( t ) ± 0 , t ) − δ 1 1 ∂ x u 0 ( s 1 ( t ) ± 0 , t ) + 1 2 ( ξ − δ 1 0 ) 2 ∂ x 2 u 0 ( s 1 ( t ) ± 0 , t ) + o ( 1 ) ,

(2.11) u s 3 ( ξ , t ) = u 3 ( s 1 ( t ) ± 0 , t ) + ( ξ − δ 1 0 ) ∂ x u 2 ( s 1 ( t ) ± 0 , t ) − δ 1 1 ∂ x u 1 ( s 1 ( t ) ± 0 , t ) + 1 2 ( ξ − δ 1 0 ) 2 ∂ x 2 u 1 ( s 1 ( t ) ± 0 , t ) − δ 1 2 ∂ x u 0 ( s 1 ( t ) ± 0 , t ) − ( ξ − δ 1 0 ) δ 1 1 ∂ x 2 u 0 ( s 1 ( t ) ± 0 , t ) + 1 6 ( ξ − δ 1 0 ) 3 ∂ x 3 u 0 ( s 1 ( t ) ± 0 , t ) + o ( 1 ) .

Similarly, near the shock x = s 2 ( t ) , we approximate u ε as

u ε ( x , t ) ∼ u ¯ s 0 ( η , t ) + ε u ¯ s 1 ( η , t ) + ε 2 u ¯ s 2 ( η , t ) + ε 3 u ¯ s 3 ( η , t ) + … ,

where

η = x − s 2 ( t ) ε + δ 2 ( t , ε ) ,

and δ 2 is a disturbance of the shock position, which is to be determined. Assume that the expansion of δ 1 ( t , ε ) is

δ 2 ( t , ε ) = δ 2 0 ( t ) + ε δ 2 1 ( t ) + ε 2 δ 2 2 ( t ) + … .

Plugging it into (1.3) and equating the coefficients of different orders of ε , we have

O 1 ε : ∂ η f ( u ¯ s 0 ) − s 2 ˙ ( t ) ∂ η u ¯ s 0 − ∂ η ( b ( u ¯ s 0 ) ∂ η u ¯ s 0 ) = 0 , O ( 1 ) : ∂ η 2 ( b ( u ¯ s 0 ) u ¯ s 1 ) − ∂ η ( f ′ ( u ¯ s 0 ) u s 1 ) + s 2 ˙ ( t ) ∂ η u ¯ s 1 = δ 2 0 ˙ ∂ η u ¯ s 0 + ∂ t u ¯ s 0 , O ( ε ) : ∂ η 2 ( b ( u ¯ s 0 ) u ¯ s 2 ( η , t ) ) − ∂ η ( f ′ ( u ¯ s 0 ) u ¯ s 2 ) + s 2 ˙ ( t ) ∂ η u ¯ s 2 = δ 2 1 ˙ ∂ η u ¯ s 0 + δ 2 0 ˙ ∂ η u ¯ s 1 + ∂ t u ¯ s 1 + 1 2 ∂ η ( f ″ ( u ¯ s 0 ) ( u ¯ s 1 ) 2 ) − 1 2 ∂ η 2 ( b ′ ( u ¯ s 0 ) ( u ¯ s 1 ) 2 ) , O ( ε 2 ) : ∂ η 2 ( b ( u ¯ s 0 ) u ¯ s 3 ) − ∂ η ( f ′ ( u ¯ s 0 ) u s 3 ) + s 2 ˙ ( t ) ∂ η u s 3 = δ 2 2 ˙ ∂ η u ¯ s 0 + δ 2 1 ˙ ∂ η u ¯ s 1 + δ 2 0 ˙ ∂ η u ¯ s 2 + ∂ t u ¯ s 2 + ∂ η ( f ″ ( u ¯ s 0 ) ( u ¯ s 1 ⋅ u ¯ s 2 ) ) + 1 6 ∂ η [ f ‴ ( u ¯ s 0 ) ( u ¯ s 1 ) 3 ] − ∂ η [ b ′ ( u ¯ s 0 ) ∂ η ( u ¯ s 1 ⋅ u ¯ s 2 ) ] − 1 6 ∂ η 2 ( b ″ ( u ¯ s 0 ) ( u ¯ s 1 ) 3 ) ,

where s 2 ˙ ( t ) = d s 2 d t , δ 2 i ˙ = d δ 2 i d t , i = 1 , 2 , 3 . Also, in the matching region, when η → ± ∞ , we have

(2.12) u ¯ s 0 ( η , t ) = u 0 ( s 2 ( t ) ± 0 , t ) + o ( 1 ) ,

(2.13) u ¯ s 1 ( η , t ) = u 1 ( s 2 ( t ) ± 0 , t ) + ( η − δ 2 0 ) ∂ x u 0 ( s 2 ( t ) ± 0 , t ) + o ( 1 ) ,

(2.14) u ¯ s 2 ( η , t ) = u 2 ( s 2 ( t ) ± 0 , t ) + ( η − δ 2 0 ) ∂ x u 1 ( s 2 ( t ) ± 0 , t ) − δ 2 1 ∂ x u 0 ( s 2 ( t ) ± 0 , t ) + 1 2 ( η − δ 2 0 ) 2 ∂ x 2 u 2 ( s 2 ( t ) ± 0 , t ) + o ( 1 ) ,

(2.15) u ¯ s 3 ( η , t ) = u 3 ( s 2 ( t ) ± 0 , t ) + ( η − δ 2 0 ) ∂ x u ¯ 2 ( s 2 ( t ) ± 0 , t ) − δ 2 1 ∂ x u ¯ 1 ( s 2 ( t ) ± 0 , t ) + 1 2 ( η − δ 2 0 ) 2 ∂ x 2 u ¯ 1 ( s 2 ( t ) ± 0 , t ) − δ 2 2 ∂ x u ¯ 0 ( s 2 ( t ) ± 0 , t ) − ( η − δ 2 0 ) δ 2 1 ∂ x 2 u ¯ 0 ( s 2 ( t ) ± 0 , t ) + 1 6 ( η − δ 2 0 ) 3 ∂ x 3 u ¯ 0 ( s 2 ( t ) ± 0 , t ) + o ( 1 ) .

Now, let us discuss the solvability of each order functions u s i , u ¯ s i with i = 0 , … , 3 in the expansions near the shocks. Note that near x = s 1 ( t ) , u s 0 ( ξ , t ) satisfies

(2.16) ∂ ξ f ( u s 0 ( ξ , t ) ) − s 1 ˙ ( t ) ∂ ξ u s 0 ( ξ , t ) = ∂ ξ ( b ( u s 0 ( ξ , t ) ) ∂ ξ u s 0 ( ξ , t ) ) ,

(2.17) u s 0 ( ξ , t ) → u l = u 0 ( s 1 ( t ) − 0 , t ) , ξ → − ∞ ,

(2.18) u s 0 ( ξ , t ) → u m = u 0 ( s 1 ( t ) + 0 , t ) , ξ → + ∞ .

Near x = s 2 ( t ) , it follows that

(2.19) ∂ η f ( u ¯ s 0 ( η , t ) ) − s 2 ˙ ( t ) ∂ η u ¯ s 0 ( η , t ) = ∂ η ( b ( u ¯ s 0 ( η , t ) ) ∂ η u ¯ s 0 ( η , t ) ) ,

(2.20) u ¯ s 0 ( η , t ) → u ¯ m = u 0 ( s 2 ( t ) − 0 , t ) , η → − ∞ ,

(2.21) u ¯ s 0 ( η , t ) → u r = u 0 ( s 2 ( t ) + 0 , t ) , η → + ∞ .

We have the following existence and properties of u s 0 and u ¯ s 0 .

Lemma 2.1

There exists a unique smooth solution u s 0 ( ξ , t ) to the boundary value problems (2.16)–(2.18), such that

(2.22) ∣ ∂ ξ u s 0 ( η , t ) ∣ ≤ C e − δ 0 ∣ ξ ∣ ,

where δ 0 > 0 is a constant. Similarly, problems (2.19)–(2.21) also have a smooth solution u ¯ s 0 ( η , t ) satisfying

(2.23) ∣ ∂ η u ¯ s 0 ( η , t ) ∣ ≤ C e − δ ¯ 0 ∣ η ∣ ,

where δ ¯ 0 > 0 is a constant.

Proof

It follows from (2.16) that

f ′ ( u s 0 ( ξ , t ) ) ∂ ξ u s 0 ( ξ , t ) − s ˙ ( t ) ∂ ξ u s 0 ( ξ , t ) = ∂ ξ b ( u s 0 ( ξ , t ) ) ∂ ξ u s 0 ( ξ , t ) + b ( u s 0 ( ξ , t ) ) ∂ ξ 2 u s 0 ( ξ , t ) .

By setting P = ∂ ξ u s 0 , we have

(2.24) ∂ ξ P = f ′ ( u s 0 ( ξ , t ) ) − s ˙ ( t ) − ∂ ξ b ( u s 0 ( ξ , t ) ) b ( u s 0 ( ξ , t ) ) P .

Integrating (2.24) from 0 to ξ shows that

ln P ( ξ ) P ( 0 ) = ∫ 0 ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) − ∂ ρ b ( u s 0 ( ρ , t ) ) b ( u s 0 ( ρ , t ) ) d ρ ,

i.e.,

(2.25) ∂ ξ u s 0 ( ξ , t ) = ∂ ξ ( u s 0 ( 0 , t ) ) b ( u l ) b ( u s 0 ( ξ , t ) ) exp ∫ 0 ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u s 0 ( ρ , t ) ) d ρ .

Again integrating (2.25) from − ∞ to + ∞ , we have

u r − u l = ∂ ξ u s 0 ( 0 , t ) b ( u l ) ∫ − ∞ + ∞ 1 b ( u s 0 ( ξ , t ) ) exp ∫ 0 ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u s 0 ( ρ , t ) ) d ρ d ξ .

Then,

∂ ξ u s 0 ( 0 , t ) b ( u l ) = u r − u l ∫ − ∞ + ∞ 1 b ( u s 0 ( ξ , t ) ) exp ∫ 0 ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u s 0 ( ρ , t ) ) d ρ d ξ .

Combining it with (2.25) gives

∂ ξ u s 0 ( ξ , t ) = ( u r − u l ) exp ∫ 0 ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u s 0 ( ρ , t ) ) d ρ b ( u s 0 ( ξ , t ) ) ∫ − ∞ + ∞ 1 b ( u s 0 ( ξ , t ) ) exp ∫ 0 ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u s 0 ( ρ , t ) ) d ρ d ξ .

Similarly, the solution u ¯ s 0 ( η , t ) of (2.19) and (2.21) satisfies

∂ η u ¯ s 0 ( η , t ) = ( u ¯ r − u ¯ l ) exp ∫ 0 η f ′ ( u ¯ s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u ¯ s 0 ( ρ , t ) ) d ρ b ( u ¯ s 0 ( η , t ) ) ∫ − ∞ + ∞ 1 b ( u s 0 ( η , t ) ) exp ∫ 0 η 1 b ( u ¯ s 0 ( η , t ) ) f ′ ( u ¯ s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u ¯ s 0 ( ρ , t ) ) d ρ d η .

Thus, the entropy condition (1.11) and the boundedness of b ( ⋅ ) imply (2.22) and (2.23).□

Next, according to (2.9), we expect u s 1 to be

u s 1 = ξ ⋅ ∂ x u 0 ( s ( t ) ± 0 , t ) + O ( 1 ) , ξ → ± ∞ .

This suggests us to write

(2.26) u s 1 ( ξ , t ) = V 1 ( η , t ) + D 1 ( ξ , t ) ,

where D 1 ( ξ , t ) is a smooth function and satisfies

D 1 ( ξ , t ) = ξ ⋅ ∂ x u 0 ( s ( t ) − 0 , t ) , ξ < − 1 , ξ ⋅ ∂ x u 0 ( s ( t ) + 0 , t ) , ξ > 1 .

It follows from (2.2) that

d d t u 0 ( s ( t ) ± 0 , t ) − ( ∂ u f ( u 0 ( s ( t ) ± 0 , t ) ) − s ˙ ( t ) ) ∂ x u 0 ( s ( t ) ± 0 , t ) = 0 .

We substitute (2.26) into (2.2) to obtain

(2.27) ∂ ξ 2 ( b ( u s 0 ( ξ , t ) ) V 1 ( ξ , t ) ) − ∂ ξ ( ∂ u f ( u s 0 ( ξ , t ) ) V 1 ( ξ , t ) ) + s ˙ ( t ) ∂ ξ V 1 ( ξ , t ) = δ ˙ 1 0 ∂ ξ u s 0 ( ξ , t ) + h ( ξ , t ) ,

where

h ( ξ , t ) = ∂ t u s 0 ( ξ , t ) − ∂ ξ 2 ( b ( u s 0 ( ξ , t ) ) D 1 ( ξ , t ) ) − s ˙ ( t ) ∂ ξ D 1 ( ξ , t ) + ∂ ξ ( ∂ u f ( u s 0 ( ξ , t ) ) D 1 ( ξ , t ) ) .

Therefore,

h ( ξ , t ) = ∂ u 2 f ( u s 0 ( ξ , t ) ) ∂ ξ u s 0 ( ξ , t ) ⋅ ξ ∂ x u 0 ( s ( t ) ± 0 , t ) + [ ∂ u f ( u s 0 ( ξ , t ) ) − s ˙ ( t ) ] ∂ x u 0 ( s ( t ) ± 0 , t ) + ∂ t u s 0 ( ξ , t ) = ∂ u 2 f ( u s 0 ( ξ , t ) ) ∂ ξ u s 0 ( ξ , t ) ⋅ ξ ∂ x u 0 ( s ( t ) ± 0 , t ) + ∫ 0 ξ ∂ t ∂ ξ u s 0 ( σ , t ) d σ − ∫ 0 ξ ∂ t ∂ ξ u s 0 ( σ , t ) d σ .

In view of Lemma 2.1, there exists a positive constant δ 0 > 0 , such that

∣ h ( ξ , t ) ∣ ≤ C e − δ 0 ∣ ξ ∣ .

Define H ( ξ , t ) = ∫ 0 ξ h ( σ , t ) d σ ; substituting it into (2.27), we obtain

(2.28) ∂ ξ [ b ( u s 0 ( ξ , t ) ) V 1 ( ξ , t ) ] = [ f ′ ( u s 0 ( ξ , t ) ) − s ˙ ( t ) ] V 1 ( ξ , t ) + δ ˙ 0 u s 0 ( ξ , t ) + H ( ξ , t ) + c ( t ) ,

where c ( t ) is a function to be determined.

Lemma 2.2

There exists a unique smooth solution V 1 ( ξ , t ) to equation (2.28) and satisfies

(2.29) V 1 ( ξ , t ) = ( s ˙ ( t ) − f ′ ( u l ) ) − 1 [ u l δ ˙ 1 0 + H − + c ( t ) ] + O ( 1 ) e − σ ∣ ξ ∣ , ξ → − ∞ , ( s ˙ ( t ) − f ′ ( u r ) ) − 1 [ u r δ ˙ 1 0 + H + + c ( t ) ] + O ( 1 ) e − σ ∣ ξ ∣ , ξ → + ∞ ,

where H ± = lim ξ → ± ∞ H ( ξ , t ) , σ > 0 .

Proof

It follows from (2.28) that

∂ ξ ( b ( u s 0 ( ξ , t ) ) V 1 ( ξ , t ) ) + s ˙ ( t ) − f ′ ( u s 0 ( ξ , t ) ) b ( u s 0 ( ξ , t ) ) b ( u s 0 ( ξ , t ) ) V 1 ( ξ , t ) = δ ˙ 1 0 u s 0 ( ξ , t ) + H ( ξ , t ) + c ( t ) ,

which is equivalent to

(2.30) ∂ ξ b ( u s 0 ( ξ , t ) ) V 1 ( ξ , t ) exp ∫ a ξ s ˙ ( t ) − f ′ ( u s 0 ( ρ , t ) ) b ( u s 0 ( ρ , t ) ) d ρ = [ δ ˙ 1 0 u s 0 ( ξ , t ) + H ( ξ , t ) + c ( t ) ] exp ∫ a ξ s ˙ ( t ) − f ′ ( u s 0 ( ρ , t ) ) b ( u s 0 ( ρ , t ) ) d ρ .

Integrating (2.30) from a to ξ shows that

b ( u s 0 ( ξ , t ) ) V 1 ( ξ , t ) exp ∫ a ξ s ˙ ( t ) − f ′ ( u s 0 ( ρ , t ) ) b ( u s 0 ( ρ , t ) ) d ρ − b ( u s 0 ( a , t ) ) V 1 ( a , t ) = ∫ a ξ [ δ ˙ 1 0 u s 0 ( τ , t ) + H ( τ , t ) + c ( t ) ] exp ∫ a τ s ˙ ( t ) − f ′ ( u s 0 ( ρ , t ) ) b ( u s 0 ( ρ , t ) ) d ρ d τ .

Then,

V 1 ( ξ , t ) = b ( u s 0 ( a , t ) ) V 1 ( a , t ) b ( u s 0 ( ξ , t ) ) exp − ∫ a ξ s ˙ ( t ) − f ′ ( u s 0 ( ρ , t ) ) b ( u s 0 ( ρ , t ) ) d ρ + 1 b ( u s 0 ( ξ , t ) ) ∫ a ξ [ δ ˙ 1 0 u s 0 ( τ , t ) + H ( τ , t ) + c ( t ) ] exp ∫ ξ τ s ˙ ( t ) − f ′ ( u s 0 ( ρ , t ) ) b ( u s 0 ( ρ , t ) ) d ρ d τ .

It follows from (1.11) that

□ lim ξ → ± ∞ ∂ ξ V 1 ( ξ , t ) ∼ O ( 1 ) exp { − α 1 ∣ ξ ∣ } , α 1 > 0 .

Now, we can select c ( t ) and δ 1 0 so that they satisfy the matching conditions (2.9) and (2.28). We obtain from (2.9) that

lim ξ → ± ∞ V 1 ( ξ , t ) = u 1 ( s ( t ) ± 0 , t ) − δ 1 0 u 0 ( s ( t ) ± 0 , t )

and gain

(2.31) [ u r δ ˙ 1 0 + H + + c ( t ) ] = [ s ˙ ( t ) − f ′ ( u r ) ] [ u 1 ( s ( t ) + 0 , t ) − δ 1 0 ∂ x u 0 ( s ( t ) + 0 , t ) ] ,

(2.32) [ u l δ ˙ 1 0 + H − + c ( t ) ] = [ s ˙ ( t ) − f ′ ( u l ) ] [ u 1 ( s ( t ) − 0 , t ) − δ 1 0 ∂ x u 0 ( s ( t ) − 0 , t ) ] .

Subtracting (2.31) from (2.32) gives

δ ˙ 1 0 ( u l − u r ) + H + − H − = [ s ˙ ( t ) − f ′ ( u l ) ] u 1 ( s ( t ) − 0 , t ) − [ s ˙ ( t ) − f ′ ( u r ) ] u 1 ( s ( t ) + 0 , t ) + δ 1 0 { [ s ˙ ( t ) − f ′ ( u r ) ] ∂ x u 0 ( s ( t ) + 0 , t ) − [ s ˙ ( t ) − f ′ ( u l ) ] ∂ x u 0 ( s ( t ) − 0 , t ) } .

Thus, we obtain a first-order linear ordinary differential equation for δ 1 0 as follows:

δ ˙ 1 0 + W 1 ( t ) δ 1 0 − W 2 ( t ) − H 1 ( t ) = 0 ,

where W 1 ( t ) , W 2 , and H 1 ( t ) are the known smooth functions. Combining with (2.31) gives the expression of δ 1 0 that

δ 1 0 ( t ) = δ 1 0 ( t ) exp − ∫ 0 t W 1 ( ρ ) d ρ + ∫ 0 t ( W 2 ( τ ) + H 1 ( τ ) exp ∫ t τ W 1 ( ρ ) d ρ ) d τ ,

which is a smooth function. Substitute δ 1 0 back into (2.31) or (2.32) to obtain c ( t ) . Again, substitute δ 1 0 and c ( t ) into (2.29) to solve V 1 ( η , t ) .

Proposition 2.1

u s 1 and δ 1 0 are the smooth functions, there exists σ > 0 such that

(2.33) u s 1 = u 1 ( s 1 ( t ) ± 0 , t ) + ( ξ − δ 1 0 ) ∂ x u 0 ( s 1 ( t ) ± 0 , t ) + O ( 1 ) e − σ ∣ ξ ∣ , ξ → ± ∞ .

And u ¯ s 1 and δ 2 0 are the smooth functions, there exists σ ¯ > 0 such that

(2.34) u ¯ s 1 = u ¯ 1 ( s 2 ( t ) ± 0 , t ) + ( η − δ 2 0 ) ∂ x u 0 ( s 2 ( t ) ± 0 , t ) + O ( 1 ) e − σ ¯ ∣ η ∣ , η → ± ∞ .

Similarly, we can determine δ 0 2 , u s 2 , u ¯ s 2 , δ 2 1 , u s 3 , δ 1 2 , u ¯ s 3 , and δ 2 2 .

Lemma 2.3

Assume that the convexity condition ∂ u 2 f ( x , t ) > 0 holds for ≤ t ≤ T , then

(2.35) ∂ ξ ∂ u f ( u s 0 ( ξ , t ) ) < 0 ,

(2.36) ∂ η ∂ u f ( u ¯ s 0 ( η , t ) ) < 0 .

Proof

Since

∂ ξ u s 0 ( ξ , t ) = ( u 0 ( s 1 ( t ) + 0 , t ) − u 0 ( s 1 ( t ) − 0 , t ) ) exp ∫ a ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u s 0 ( ρ , t ) ) d ρ b ( u s 0 ( ξ , t ) ) ∫ − ∞ + ∞ 1 b ( u s 0 ( ξ , t ) ) exp ∫ a ξ f ′ ( u s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u s 0 ( ρ , t ) ) d ρ d ξ , ∂ η u ¯ s 0 ( η , t ) = ( u 0 ( s 2 ( t ) + 0 , t ) − u 0 ( s 2 ( t ) − 0 , t ) ) exp ∫ a η f ′ ( u ¯ s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u ¯ s 0 ( ρ , t ) ) d ρ b ( u ¯ s 0 ( η , t ) ) ∫ − ∞ + ∞ 1 b ( u s 0 ( η , t ) ) exp ∫ a η f ′ ( u ¯ s 0 ( ρ , t ) ) − s ˙ ( t ) b ( u ¯ s 0 ( ρ , t ) ) d ρ d η .

Then, it follows from the entropy condition (1.11) that ∂ ξ u s 0 < 0 and ∂ η u ¯ s 0 < 0 . Furthermore, we have

∂ ξ ∂ u f ( u s 0 ( ξ , t ) ) = ∂ u 2 f ( u s 0 ( ξ , t ) ) ∂ ξ u s 0 ( ξ , t ) ,

□ ∂ η ∂ u f ( u ¯ s 0 ( η , t ) ) = ∂ u 2 f ( u ¯ s 0 ( η , t ) ) ∂ η u ¯ s 0 ( η , t ) .

2.3 Approximate solutions

We now construct an approximate solution to (1.3) by patching the truncated outer and shock-layer solutions as

O ( x , t ) = u 0 ( x , t ) + ε u 1 ( x , t ) + ε 2 u 2 ( x , t ) + ε 3 u 3 ( x , t ) , x ≠ s i ( t ) , i = 1 , 2 , I 1 ( x , t ) = u s 0 x − s 1 ( t ) ε + δ 1 0 + ε δ 1 1 + ε 2 δ 1 2 , t + ε u s 1 x − s 1 ( t ) ε + δ 1 0 + ε δ 1 1 + ε 2 δ 1 2 , t

+ ε 2 u s 2 x − s 1 ( t ) ε + δ 1 0 + ε δ 1 1 + ε 2 δ 1 2 , t + ε 3 u s 3 x − s 1 ( t ) ε + δ 1 0 + ε δ 1 1 + ε 2 δ 1 2 , t , I 2 ( x , t ) = u ¯ s 0 x − s 2 ( t ) ε + δ 2 0 + ε δ 2 1 + ε 2 δ 2 2 , t + ε u ¯ s 1 x − s 2 ( t ) ε + δ 2 0 + ε δ 2 1 + ε 2 δ 2 2 , t + ε 2 u ¯ s 2 x − s 2 ( t ) ε + δ 2 0 + ε δ 2 1 + ε 2 δ 2 2 , t + ε 3 u ¯ s 3 x − s 2 ( t ) ε + δ 2 0 + ε δ 2 1 + ε 2 δ 2 2 , t .

The functions u i , u s i , u ¯ s i , δ 1 i , and δ 2 i , i = 0 , 1 , 2 , 3 were given in the previous section. Then, the approximate solution to (1.3) is defined as

(2.37) u ¯ a ( x , t ) = m 1 I 1 + m 2 I 2 + ( 1 − m 1 − m 2 ) O + d ( x , t ) = u a ( x , t ) + d ( x , t ) ,

where m i = m ( x − s i ( t ) ε γ ) , i = 1 , 2 , γ ∈ ( 6 7 , 1 ) with m being defined in (1.12), and d ( x , t ) is a higher-order correction term to be determined. Due to the structure of the various orders of inner and outer solutions, u ¯ a solves

(2.38) ∂ t u ¯ a + ∂ x f ( u ¯ a ) − ε ∂ x ( b ( u ¯ a ) ∂ x u ¯ a ) = ∑ i = 1 5 q i ( x , t ) ,

(2.39) u ¯ a ( x , t = 0 ) = m 1 ∑ i = 0 3 ε i u s i ( ξ , 0 ) + m 2 ∑ i = 0 3 ε i u ¯ s i ( η , 0 ) + ( 1 − m 1 − m 2 ) ∑ i = 0 3 ε i u 0 i ( x ) ,

where q i ( x , t ) are the smooth functions as follows:

q 1 ( x , t ) = ( 1 − m 1 − m 2 ) f ( O ) − f ( u 0 ) − ε f ′ ( u 0 ) u 1 − ε 2 f ′ ( u 0 ) u 2 − ε 2 2 f ″ ( u 0 ) ( u 1 ) 2 − ε 3 f ″ ( u 0 ) ( u 1 ⋅ u 2 ) − ε 3 6 f ‴ ( u 0 ) ( u 1 ) 3 − ε 4 f ″ ( u 0 ) ( u 1 ⋅ u 3 ) − ε 4 2 f ″ ( u 0 ) ( u 2 ) 2 − ε 4 2 f ‴ ( u 0 ) ( ( u 1 ) 2 ⋅ u 2 ) − ε 4 24 f ( 4 ) ( u 0 ) ( u 1 ) 4 x − ε 5 ∂ x 2 ( b ( u 0 ) u 4 ) − ε 5 ∂ x ( b ′ ( u 0 ) ∂ x ( u 1 ⋅ u 3 ) ) + 1 2 ∂ x ( b ″ ( u 0 ) ( u 1 ) 2 ∂ x u 2 ) + 1 2 ∂ x 2 ( b ′ ( u 0 ) ( u 2 ) 2 ) + 1 24 ∂ x 2 ( b ‴ ( u 0 ) ( u 1 ) 4 ) , q 2 ( x , t ) = m 1 f ( I 1 ) − f ( u s 0 ) − ε f ′ ( u s 0 ) u s 1 − ε 2 f ′ ( u s 0 ) u s 2 − ε 3 f ′ ( u s 0 ) u s 3 − ε 2 2 f ′ ( u s 0 ) ( u s 1 ) 2 − ε 3 f ″ ( u s 0 ) ( u 1 1 ⋅ u s 2 ) − ε 3 6 f ‴ ( u s 0 ) ( u s 1 ) 3 x + ε 3 ∂ t u s 3 + ε 4 ( δ ˙ 1 0 u s 3 + δ ˙ 1 1 u s 2 + δ ˙ 1 2 u s 1 + δ ˙ 1 3 u s 0 − s ˙ 1 u s 3 + ε δ 1 2 u s 2 ) x , q 3 ( x , t ) = m 2 f ( I 2 ) − f ( u ¯ s 0 ) − ε f ′ ( u ¯ s 0 ) u ¯ s 1 − ε 2 f ′ ( u ¯ s 0 ) u ¯ s 2 − ε 3 f ′ ( u ¯ s 0 ) u ¯ s 3 − ε 2 2 f ′ ( u ¯ s 0 ) ( u ¯ s 1 ) 2 − ε 3 f ″ ( u ¯ s 0 ) ( u ¯ 1 1 ⋅ u ¯ s 2 ) − ε 3 6 f ‴ ( u ¯ s 0 ) ( u ¯ s 1 ) 3 x + ε 3 ∂ t u ¯ s 3 + ε 4 ( δ ˙ 2 0 u ¯ s 3 + δ ˙ 2 1 u ¯ s 2 + δ ˙ 2 2 u ¯ s 1 + δ ˙ 2 3 u ¯ s 0 − s ˙ 2 u ¯ s 3 + ε δ 2 2 u ¯ s 2 ) x , q 4 ( x , t ) = ∂ t m 1 ( I 1 − O ) + ∂ t m 2 ( I 2 − O ) + f ( m 1 I 1 + m 2 I 2 + ( 1 − m 1 − m 2 ) O ) x − { m 1 f ( I 1 ) + m 2 f ( I 2 ) + ( 1 − m 1 − m 2 ) f ( O ) } x + ε ( b ( O ) O x ) x + ∂ x m 1 ( f ( I 1 ) − f ( O ) ) + ∂ x m 2 ( f ( I 2 ) − f ( O ) ) + ε { m 1 ( b ( I 1 ) − b ( O ) ) ( I 1 − O ) x + m 2 ( b ( I 2 ) − b ( O ) ) ( I 2 − O ) x } x − ε ∂ x m 1 ( b ( I 1 ) − b ( O ) ) ( I 1 − O ) x − ε ∂ x m 2 ( b ( I 2 ) − b ( O ) ) ( I 2 − O ) x + ε { m 1 b ( O ) ( I 1 − O ) x + m 2 b ( O ) ( I 2 − O ) x } x − ε ∂ x m 1 b ( O ) ( I 1 − O ) x − ε ∂ x m 2 b ( O ) ( I 2 − O ) x + ε { m 1 ( b ( I 1 ) − b ( O ) ) O x + m 2 ( b ( I 2 ) − b ( O ) ) O x } x − ε ∂ x m 1 [ b ( I 1 ) − b ( O ) ] O x − ε ∂ x m 2 [ b ( I 2 ) − b ( O ) ] O x − ε { g ( m 1 I 1 + m 2 I 2 + ( 1 − m 1 − m 2 ) O ) } x x , q 5 ( x , t ) = d t − ε ( g ( u ¯ a ) − g ( u ¯ a − d ) ) x x + ( f ( u ¯ a ) − f ( u ¯ a − d ) ) x .

In view of our construction, we have (i) supp q 1 ⊆ { ( x , t ) : s i ( t ) + ε γ < x or x < s i ( t ) − ε γ , 0 ≤ t ≤ T , i = 1 , 2 } ,

(2.40) ∂ x k q 1 ( x , t ) = O ( 1 ) ε 5 − k γ , ∫ 0 T ( ‖ ∂ x k q 1 ( ⋅ , t ) ‖ 2 d t ) 1 2 ≤ O ( 1 ) ε 5 − ( k − 1 2 ) γ , k = 0 , 1 , 2 ,

(ii) supp q 2 ⊆ { ( x , t ) : s 1 ( t ) − 2 ε γ < x < s 1 ( t ) + 2 ε γ , 0 ≤ t ≤ T } ,

(2.41) ∂ x k q 2 ( x , t ) = O ( 1 ) ε ( 3 − l ) γ , k = 0 , 1 , 2 ,

(iii) supp q 3 ⊆ { ( x , t ) : s 1 ( t ) − 2 ε γ < x < s 2 ( t ) + 2 ε γ , 0 ≤ t ≤ T } ,

(2.42) ∂ x k q 3 ( x , t ) = O ( 1 ) ε ( 3 − l ) γ , k = 0 , 1 , 2 ,

(iv) supp q 4 ⊆ { ( x , t ) : ε γ ≤ ∣ x − s i ( t ) ∣ ≤ 2 ε γ , 0 ≤ t ≤ T } ,

(2.43) ∂ x k q 4 ( x , t ) = O ( 1 ) ε ( 3 − l ) γ , k = 0 , 1 , 2 .

Here, we have used the estimate

(2.44) ∂ x l ( I 1 − O ) = O ( 1 ) ε ( 4 − l ) γ , l = 0 , 1 , 2 ,

on { ( x , t ) : ε γ ≤ ∣ x − s 1 ( t ) ∣ ≤ 2 ε γ , 0 ≤ t ≤ T } . Similarly,

(2.45) ∂ x l ( I 2 − O ) = O ( 1 ) ε ( 4 − l ) γ , l = 0 , 1 , 2 ,

on { ( x , t ) : ε γ ≤ ∣ x − s 2 ( t ) ∣ ≤ 2 ε γ , 0 ≤ t ≤ T } , which can be obtained by the matching conditions (2.8)–(2.11), (2.12)–(2.15), and O ( 1 ) = exp { − α 0 ∣ ξ ∣ } . Set R ε = ∑ i = 1 4 q i ( x , t ) , then R ε = O ( 1 ) ε 3 γ . We now choose d ( x , t ) to be the solution of the diffusion problem

(2.46) d t = ε ( b ( u a ) d x ) x − ∑ i = 1 4 q i ( x , t ) , d ( x , 0 ) = 0 .

Consequently, for u ¯ a , we obtain the conservative form

(2.47) ∂ t u ¯ a + ∂ f ( u ¯ a ) − ε ∂ x ( b ( u ¯ a ) ∂ x u ¯ a ) = ε ( b ( u a ) d x ) x − ε [ g ( u ¯ a ) − g ( u a ) ] x x + ( f ( u ¯ a ) − f ( u a ) ) x ,

with the initial data (2.39). It then remains to estimate the linear diffusion wave d ( x , t ) .

Lemma 2.4

Let d ( x , t ) be the solution of (2.46). Then, the following estimates

(2.48) sup 0 ≤ t ≤ T ‖ d ‖ L 2 ( R ) 2 + ε ∫ 0 T ‖ ∂ x d ‖ L 2 ( R ) 2 d t ≤ C ε 7 γ ,

(2.49) sup 0 ≤ t ≤ T ‖ ∂ x d ‖ L 2 ( R ) 2 + ε ∫ 0 T ‖ ∂ x 2 d ‖ L 2 ( R ) 2 d t ≤ C ε 7 γ − 2 ,

(2.50) sup 0 ≤ t ≤ T ‖ ∂ x 2 d ‖ L 2 ( R ) 2 + ε ∫ 0 T ‖ ∂ x 3 d ‖ L 2 ( R ) 2 d t ≤ C ε 7 γ − 4 ,

(2.51) sup 0 ≤ t ≤ T ‖ d ( x , t ) ‖ L ∞ ( R ) ≤ C ε 7 γ − 1 2 ,

(2.52) sup 0 ≤ t ≤ T ‖ ∂ x d ( x , t ) ‖ L ∞ ( R ) ≤ C ε 7 γ − 3 2 ,

hold for all t ∈ [ 0 , T ] .

Proof

Multiplying (2.46) by d , integrating on R , and using integration by parts, we obtain

1 2 d d t ∫ R d 2 d x + ε ∫ R b ( u a ) ∣ ∂ x d ∣ 2 d x + ∫ R d ⋅ R ε d x = 0 .

By the uniform parabolic condition and Gronwall’s inequality, we obtain (2.48). Let D = ∂ x d , and D satisfies

(2.53) D t = ε ( b ( u a ) D ) x x − ∂ x R ε , D ( x , 0 ) = 0 .

Then,

1 2 d d t ∫ R D 2 d x + ( C 0 − β ) ε ∫ R ∣ ∂ x D ∣ 2 d x ≤ ε − 1 ∫ R D 2 d x + C ∫ R D 2 d x + C ε 5 γ ,

where β is small enough satisfying C 0 − β > 0 . Then, it is easy to obtain (2.49) after integration over [ 0 , T ] . Next, let D ¯ = ∂ x D , and D ¯ satisfies

(2.54) D ¯ t = ε ( b ( u a ) D ) x x x − ∂ x 2 R ε , D ¯ ( x , 0 ) = 0 .

By the same procedure, we obtain (2.49). It follows from (2.48) and (2.49) that

(2.55) sup 0 ≤ t ≤ T ‖ d ( x , t ) ‖ L ∞ ( R ) ≤ 2 ‖ d ( x , t ) ‖ L R 2 1 2 ‖ ∂ x d ( x , t ) ‖ L R 2 1 2 ≤ C ε 7 γ − 1 2 .

Similarly, (2.52) follows from (2.49) and (2.50).□

Lemma 2.5

Let u ¯ a ( x , t ) be defined as in (2.37). Then,

(2.56) u ¯ a ( x , t ) = u 0 ( x , t ) + O ( 1 ) ε , ∣ x − s i ( t ) ∣ ≥ ε γ , i = 1 , 2 , u s 0 ( ξ , t ) + O ( 1 ) ε γ , ∣ x − s 1 ( t ) ∣ ≤ 2 ε γ , u ¯ s 0 ( η , t ) + O ( 1 ) ε γ , ∣ x − s 2 ( t ) ∣ ≤ 2 ε γ .

Proof

According to our construction process, we have

u ¯ a ( x , t ) = I 1 + d , ∣ x − s 1 ( t ) ∣ ≤ ε γ , O + m 1 ( I 1 − O ) + d , ε γ ≤ ∣ x − s 1 ( t ) ∣ ≤ 2 ε γ , O + d , ∣ x − s i ( t ) ∣ ≥ 2 ε γ , i = 1 , 2 , O + m 2 ( I 2 − O ) + d , ε γ ≤ ∣ x − s 2 ( t ) ∣ ≤ 2 ε γ , I 2 + d , ∣ x − s 2 ( t ) ∣ ≤ ε γ ,

where O ( x , t ) = u 0 ( x , t ) + O ( 1 ) ε on ∣ x − s i ( t ) ∣ > ε γ , i = 1 , 2 , and I 1 ( x , t ) = u s 0 ( ξ , t ) + O ( 1 ) ε γ on s 1 ( t ) − 2 ε γ < x < s 1 ( t ) + 2 ε γ . Similarly, I 2 ( x , t ) = u ¯ s 0 ( η , t ) + O ( 1 ) ε γ on s 2 ( t ) − 2 ε γ < x < s 2 ( t ) + 2 ε γ . These, together with the structure of the approximate solution, result in (2.56).□

3 Vanishing viscosity limit

In this section, we prove that there exists an exact solution of (1.3) in the vicinity of the approximate solution constructed and establish the asymptotic equivalence between the two viscous equations and the inviscid equation. We first derive the error equation, and using the structure of the approximate solution constructed in the previous section to obtain the H 1 -estimates on the error equation, which consequently implies the vanishing viscosity limit away from the shocks.

3.1 Error equation

We decompose the exact solution u ε ( x , t ) as the sum of the approximate solution u ¯ a ( x , t ) and the error term

u ε ( x , t ) = u ¯ a ( x , t ) + ε 1 ⁄ 2 + δ v ( x , t ) , δ ∈ 0 , 1 2 , x ∈ R , t ∈ [ 0 , T ] .

It follows from (1.13), (2.39), and (2.47) that

(3.1) ∂ t v − ε 1 ⁄ 2 − δ ∂ x 2 [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] + ε 1 ⁄ 2 − δ ∂ x [ b ( u ¯ a − d ) ∂ x d ] + ε − ( 1 ⁄ 2 + δ ) ∂ x [ f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ] = 0 ,

v ( x , 0 ) = O ( ε 2 ) .

To exploit the compressibility of the shocks, we set

φ ( x , t ) = ∫ − ∞ x v ( z , t ) d z , ∀ x ∈ R .

Then, together with (1.13), it follows

(3.2) ∂ t φ − ε 1 ⁄ 2 − δ ∂ x [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] + ε 1 ⁄ 2 − δ b ( u ¯ a − d ) ∂ x d + ε − ( 1 ⁄ 2 + δ ) [ f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ] = 0 .

(3.3) φ ( x , 0 ) = O ( ε 2 ) .

Our purpose is to show the Cauchy problems (3.2)–(3.3) have a unique solution v ∈ C 1 ( [ 0 , T ] ; H 2 ( R ) ) with the property that

(3.4) ‖ v ‖ L ∞ ( [ 0 , T ] × R ) ≤ C ε 7 γ ⁄ 2 − δ − 2 .

So (3.4) is a consequence of the following priori estimates.

Proposition 3.1

Suppose that for ∀ ε > 0 , there exists a unique solution φ ∈ C 1 ( [ 0 , T ] ; H 2 ( R ) ) to the Cauchy problems (3.2)–(3.3). There exist positive constants C and γ , which are independent of ε , such that if

(3.5) sup 0 ≤ t ≤ T ‖ ∂ x φ ‖ L ∞ ( R ) ≤ C ,

then

(3.6) sup 0 ≤ t ≤ T ‖ φ ‖ H 2 ( R ) 2 + ε ∫ 0 T ‖ ∂ x φ ‖ H 2 ( R ) 2 d t ≤ C ε 7 γ − 2 δ − 5 ,

where 6 7 < γ < 1 .

To prove this proposition, we need the H 2 -estimate of the error term φ , which is presented in the following three subsections.

3.2 Basic L 2 estimate

Lemma 3.1

Under the assumptions of Proposition 3.1, there exists a positive constant C such that φ ∈ C 1 ( [ 0 , T ] ; H 2 ( R ) ) is a solution to problems (3.2)–(3.3), satisfying

(3.7) sup 0 ≤ t ≤ T ‖ φ ‖ L 2 ( R ) 2 + ε ∫ 0 T ‖ ∂ x φ ‖ L 2 ( R ) 2 d t ≤ C ε 7 γ − 2 δ − 1 .

Proof

Multiplying (3.2) by φ and integrating over R yield after integration by parts that

1 2 d d t ∫ R ∣ φ ∣ 2 d x + H + I + J = 0 ,

where

H = − ε 1 ⁄ 2 − δ ∫ R ∂ x [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] ⋅ φ d x , I = ε − ( 1 ⁄ 2 + δ ) ∫ R [ f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ] ⋅ φ d x , J = ε 1 ⁄ 2 − δ ∫ R b ( u ¯ a − d ) ∂ x d ⋅ φ d x .

In view of the structure of u ¯ a , we have

H = ε 1 ⁄ 2 − δ ∫ R [ g ′ ( u a ) ( ε 1 ⁄ 2 + δ φ x + d ) + O ( 1 ) ( ε 1 ⁄ 2 + δ φ x + d ) 2 ] ⋅ φ x d x = ε ∫ R b ( u a ) ∣ φ x ∣ 2 d x + ε 1 ⁄ 2 − δ ∫ R b ( u a ) d ⋅ φ x d x + ε 1 ⁄ 2 − δ ∫ R O ( 1 ) ( ε 1 ⁄ 2 + δ φ x + d ) 2 ⋅ φ x d x = ε ∫ R b ( u a ) ∣ φ x ∣ 2 d x + A 1 + A 2 .

It follows from the uniform parabolic condition that there exists C 0 > 0 such that

C 0 ε ∫ R ∣ φ x ∣ 2 d x ≤ ε ∫ R b ( u a ) ∣ φ x ∣ 2 d x .

Based on the previous estimation for ‖ d ‖ L 2 ( R ) 2 , we have

∣ A 1 ∣ = ∣ ε 1 ⁄ 2 − δ ∫ R b ( u a ) d ⋅ φ x d x ∣ ≤ C ε 1 ⁄ 2 − δ ∫ R ∣ d ⋅ φ x ∣ d x ≤ C ε − 2 δ ∫ R ∣ d ∣ 2 d x + β ε ∫ R ∣ φ x ∣ 2 d x ≤ C ε 7 γ − 2 δ − 1 + β ε ∫ R ∣ φ x ∣ 2 d x .

In view of estimate (2.51), there exist positive constants β 1 and β 2 such that

∣ A 2 ∣ = ∣ ε 1 ⁄ 2 − δ ∫ R O ( 1 ) ( ε 1 ⁄ 2 + δ φ x + d ) 2 ⋅ φ x d x ∣ ≤ C ε 1 ⁄ 2 − δ ∫ R ∣ d 2 ⋅ φ x ∣ d x + C ε 3 ⁄ 2 + δ ∫ R ∣ ( φ x ) 2 ⋅ φ x ∣ d x ≤ C ε 7 γ ⁄ 2 − δ ∫ R ∣ d ∣ 2 d x + β 1 ε ∫ R ∣ φ x ∣ 2 d x + β 2 ε ∫ R ∣ φ x ∣ 2 d x ≤ C ε 21 γ ⁄ 2 − δ + ( β 1 + β 2 ) ε ∫ R ∣ φ x ∣ 2 d x ,

where we have used assumption (3.5). Next, we have

I = ε − ( 1 ⁄ 2 + δ ) ∫ R ( ∂ u f ( u a ) ( ε 1 ⁄ 2 + δ φ x + d ) + O ( 1 ) ( ε 1 ⁄ 2 + δ φ x + d ) 2 ) ⋅ φ d x = ε − ( 1 ⁄ 2 + δ ) ∫ R ∂ u f ( u a ) d ⋅ φ d x + ∫ R ∂ u f ( u a ) φ x ⋅ φ d x + O ( 1 ) ε − ( 1 ⁄ 2 + δ ) ∫ R ( ε 1 ⁄ 2 + δ φ x + d ) 2 ⋅ φ d x = ∑ i = 1 3 B i ,

where

B 1 = ε − ( 1 ⁄ 2 + δ ) ∫ R ∂ u f ( u a ) d ⋅ φ d x ≤ C ε − 1 − 2 δ ∫ R ∣ d ∣ 2 d x + C ∫ R ∣ φ ∣ 2 d x ≤ C ∫ R ∣ φ ∣ 2 d x + C ε 7 γ − 2 δ − 1 .

In view of the structure of the approximate solutions, we obtain

B 2 = ∫ R ∂ u f ( u a ) φ ⋅ ∂ x φ d x = 1 2 ∫ R ∂ u f ( u a ) ∂ x φ 2 d x = 1 2 ∫ − ∞ s 1 ( t ) − 2 ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 1 ( t ) − 2 ε γ s 1 ( t ) − ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 1 ( t ) − ε γ s 1 ( t ) + ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 1 ( t ) + ε γ s 1 ( t ) + 2 ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 1 ( t ) + 2 ε γ s 2 ( t ) − 2 ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 2 ( t ) − 2 ε γ s 2 ( t ) − ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 2 ( t ) − ε γ s 2 ( t ) + ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 2 ( t ) + ε γ s 2 ( t ) + 2 ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x + 1 2 ∫ s 2 ( t ) + 2 ε γ + ∞ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x = ∑ i = 1 9 K i ,

with

K 1 = 1 2 ∫ − ∞ s 1 ( t ) − 2 ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x = − 1 2 ∫ − ∞ s 1 ( t ) − 2 ε γ ∂ x ∂ u f ( u a ) ⋅ φ 2 d x = O ( 1 ) ∫ − ∞ s 1 ( t ) − 2 ε γ φ 2 d x ,

and

K 2 = 1 2 ∫ s 1 ( t ) − 2 ε γ s 1 ( t ) − ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x = − 1 2 ∫ s 1 ( t ) − 2 ε γ s 1 ( t ) − ε γ ∂ u 2 f ( u a ) ( ∂ x m 1 ( I 1 − O ) + m 1 ∂ x ( I 1 − O ) + ∂ x O ) ⋅ φ 2 d x = O ( 1 ) ∫ s 1 ( t ) − 2 ε γ s 1 ( t ) − ε γ φ 2 d x ,

where we have used

∂ x l ( I 1 − O ) = O ( 1 ) ε ( 4 − l ) γ .

It follows from (2.35) that

K 3 = 1 2 ∫ s 1 ( t ) − ε γ s 1 ( t ) + ε γ ∂ u f ( u a ) ⋅ ∂ x φ 2 d x = − 1 2 ∫ s 1 ( t ) − ε γ s 1 ( t ) + ε γ 1 ε ∂ ξ ∂ u f ( I 1 ) ⋅ φ 2 d x = 1 2 ε ∫ s 1 ( t ) − ε γ s 1 ( t ) + ε γ ∣ ∂ ξ ∂ u f ( u s 0 ) ⋅ φ 2 ∣ d x + O ( 1 ) ∫ s 1 ( t ) − ε γ s 1 ( t ) + ε γ φ 2 d x ,

where we have used (2.35). Similarly, we have

K 4 + K 5 + K 6 = O ( 1 ) ∫ s 1 ( t ) + ε γ s 1 ( t ) + 2 ε γ φ 2 d x + ∫ s 1 ( t ) + 2 ε γ s 2 ( t ) − 2 ε γ φ 2 d x + ∫ s 2 ( t ) − 2 ε γ s 2 ( t ) − ε γ φ 2 d x ,

K 7 = 1 2 ε ∫ s 2 ( t ) − ε γ s 2 ( t ) + ε γ ∣ ∂ η ∂ u f ( u ¯ s 0 ) φ 2 ∣ d x + O ( 1 ) ∫ s 2 ( t ) − ε γ s 2 ( t ) + ε γ φ 2 d x , K 8 + K 9 = O ( 1 ) ∫ s 2 ( t ) + ε γ s 2 ( t ) + 2 ε γ φ 2 d x + ∫ s 2 ( t ) + 2 ε γ + ∞ φ 2 d x .

Then, in view of (2.51) and (3.3), the following estimates are obtained:

B 3 ≤ C ε − ( 1 ⁄ 2 + δ ) ∫ R ∣ d 2 ⋅ φ ∣ d x + ε 1 + 2 δ ∣ ( φ x ) 2 ⋅ φ ∣ d x ≤ C ε 7 γ ⁄ 2 − 1 − δ ∫ R ∣ d ∣ 2 d x + C ε 2 δ ⋅ ε ∫ R ∣ φ x ∣ 2 d x + C ∫ R ∣ φ ∣ 2 d x ≤ C ε 21 γ ⁄ 2 − δ − 1 + β 3 ε ∫ R ∣ φ x ∣ 2 d x + C ∫ R ∣ φ ∣ 2 d x ,

for some β 3 > 0 . Then, combining (2.49), we have

∣ J ∣ ≤ ε 1 ⁄ 2 − δ ∫ R ∣ b ( u a ) ∂ x d ⋅ φ ∣ d x ≤ C ε 1 − 2 δ ∫ R ∣ d x ∣ 2 d x + C ∫ R ∣ φ ∣ 2 d x .

Then, collecting all the estimates to obtain after choosing β i , i = 1 , 2 , 3 small enough that

1 2 d d t ∫ R φ 2 d x + ε ∫ R ∣ ∂ x φ ∣ d x + 1 2 ε ∫ s 1 ( t ) − ε γ s 1 ( t ) + ε γ ∣ ∂ ξ ∂ u f ( u s 0 ( x , t ) ) φ 2 ∣ d x + 1 2 ε ∫ s 2 ( t ) − ε γ s 2 ( t ) + ε γ ∣ ∂ η ∂ u f ( u ¯ s 0 ) φ 2 ∣ d x ≤ C ∫ R φ 2 d x + C ε 7 γ − 2 δ − 1 .

Then, Gronwall’s inequality implies (3.7). Thus, the proof of Lemma 3.1 is complete.□

3.3 Estimate of the first-order derivative

Lemma 3.2

Under the same assumptions as in Proposition 3.1, there is a positive constant C such that

(3.8) sup 0 ≤ t ≤ T ‖ ∂ x φ ‖ L 2 ( R ) 2 + ε ∫ 0 T ‖ ∂ x 2 φ ‖ L 2 ( R ) 2 d t ≤ C ε 7 γ − 2 δ − 3 .

Proof

Multiplying (3.1) by v and integrating over R yield after integration by parts that

(3.9) 1 2 d d t ∫ R ∣ v ∣ 2 d x − ε 1 ⁄ 2 − δ ∫ R ∂ x 2 [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] v d x + ε − ( 1 ⁄ 2 + δ ) ∫ R ∂ x [ f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ] v d x + ε 1 ⁄ 2 − δ ∫ R ∂ x [ b ( u ¯ a − d ) ∂ x d ] v d x = 0 .

The same estimate as in Lemma 3.1, the second term can be written as

− ε 1 ⁄ 2 − δ ∫ R ∂ x 2 [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] ⋅ v d x = ε 1 ⁄ 2 − δ ∫ R g ″ ( u ξ ) ∂ x u ξ ( ε 1 ⁄ 2 + δ v + d ) ⋅ ∂ x v d x + ε ∫ R g ′ ( u ξ ) ∣ ∂ x v ∣ 2 d x + ε 1 ⁄ 2 − δ ∫ R g ′ ( u ξ ) ∂ x d ⋅ ∂ x v d x = D 1 + ε ∫ R g ′ ( u ξ ) ∣ ∂ x v ∣ 2 d x + D 2 ,

where u ξ ∈ ( u ¯ a − d , u ¯ a + ε 5 ⁄ 8 v ) , and

D 1 = ε 1 ⁄ 2 − δ ∫ R [ g ″ ( u ξ ) ∂ x u ξ ( ε 1 ⁄ 2 + δ v + d ) ] ⋅ v x d x ≤ O ( 1 ) ε ∫ R 1 ε ⋅ v ⋅ v x d x + O ( 1 ) ε 1 ⁄ 2 − δ ∫ R 1 ε ⋅ d ⋅ v x d x ≤ C ε − 1 ∫ R ∣ v ∣ 2 d x + β ε ∫ R ∣ v x ∣ 2 d x + C ε − 2 δ ∫ R ∣ d ∣ 2 d x

and

D 2 ≤ C ε 1 ⁄ 2 − δ ∫ R d x ⋅ v x d x ≤ C ε − 2 δ ∫ R ∣ d x ∣ 2 d x + β ε ∫ R ∣ v x ∣ 2 d x .

By (2.37), we obtain

ε − ( 1 ⁄ 2 + δ ) ∣ ∫ R ( f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ) ⋅ ∂ x v d x ∣ ≤ O ( 1 ) ε − ( 1 ⁄ 2 + δ ) ∫ R ∣ ∂ u f ( u ξ ) d ⋅ ∂ x v ∣ d x + ∫ R ∣ ∂ u f ( u ξ ) v ⋅ ∂ x v ∣ d x ≤ C ε − 2 − 2 δ ∫ R ∣ d ∣ 2 d x + 2 β ε ∫ R ∣ ∂ x v ∣ 2 d x + C ε − 1 ∫ R ∣ v ∣ 2 d x ≤ C ε 7 γ − 2 δ − 2 + 2 β ε ∫ R ∣ ∂ x v ∣ 2 d x + C ε − 1 ∫ R ∣ v ∣ 2 d x ,

where u ξ ∈ ( u ¯ a − d , u ¯ a + ε 5 ⁄ 8 v ) , which is probably different from that mentioned earlier, but for simplicity of notations, we still denote it by u ξ , and we will not mention it later. Next, we turn to estimate the fourth term of (3.9), and we have

ε 1 ⁄ 2 − δ ∫ R ∂ x [ b ( u ¯ a − d ) ∂ x d ] ⋅ v d x = − ε 1 ⁄ 2 − δ ∫ R b ( u a ) ∂ x d ⋅ ∂ x v d x ≤ C ε − 2 δ ∫ R ∣ ∂ x d ∣ 2 d x + β ε ∫ R ∣ ∂ x v ∣ 2 d x .

Combining all the aforementioned estimates, and choosing β such that C 0 − 4 β > 0 yield

1 2 d d t ∫ R ∣ v ∣ 2 d x + ( C 0 − 4 β ) ε ∫ R ∣ ∂ x v ∣ 2 d x ≤ C ε 7 γ − 2 δ − 2 + C ε − 2 δ ∫ R ∣ d x ∣ 2 d x + C ε − 1 ∫ R ∣ v ∣ 2 d x ≤ C ε 7 γ − 2 δ − 2 + C ε 7 γ − 2 δ − 2 + C ε − 1 ∫ R ∣ v ∣ 2 d x .

Gronwall’s inequality implies that

(3.10) sup 0 ≤ t ≤ T ∫ R ∣ v ∣ 2 d x + ε ∫ 0 T ∫ R ∣ ∂ x v ∣ 2 d x ≤ C ε 7 γ − 2 δ − 3 ,

which is exactly (3.8).□

3.4 Estimate of the second derivative

To give the L ∞ bound of the error term ∂ x φ , we still need to estimate the second derivative ∂ x 2 φ .

Lemma 3.3

Under the same assumptions as in Proposition 3.1, there is a positive constant C such that

(3.11) sup 0 ≤ t ≤ T ‖ ∂ x 2 φ ‖ L 2 ( R ) 2 + ε ∫ 0 T ‖ ∂ x 3 φ ‖ L 2 ( R ) 2 d t ≤ C ε 7 γ − 2 δ − 5 .

Proof

Let θ ( x , t ) = ∂ x v ( x , t ) = ∂ x 2 φ ( x , t ) , then θ satisfies

(3.12) ∂ t θ − ε 1 ⁄ 2 − δ ∂ x 3 [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] + ε − ( 1 ⁄ 2 + δ ) ∂ x 2 [ f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ] + ε 1 ⁄ 2 − δ ∂ x 2 [ b ( u ¯ a − d ) ∂ x d ] = 0 .

Multiplying (3.12) by θ and integrating over R yield after integration by parts that

(3.13) 1 2 d d t ∫ R ∣ θ ∣ 2 d x − ε 1 ⁄ 2 − δ ∫ R ∂ x 3 [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] θ d x + ε − ( 1 ⁄ 2 + δ ) ∫ R ∂ x 2 [ f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ] θ d x + ε 1 ⁄ 2 − δ ∫ R ∂ x 2 [ b ( u ¯ a − d ) ∂ x d ] θ d x = 0 ,

with

θ ( x , 0 ) = O ( ε ) .

First, we estimate the second term of (3.13), which is written as

− ε 1 ⁄ 2 − δ ∫ R ∂ x 3 [ g ( u ¯ a + ε 1 ⁄ 2 + δ v ) − g ( u ¯ a − d ) ] ⋅ θ d x = ε 1 ⁄ 2 − δ ∫ R [ ∂ x 2 g ′ ( u ξ ) ( ε 1 ⁄ 2 + δ v + d ) + 2 ∂ x g ′ ( u ξ ) ( ε 1 ⁄ 2 + δ ∂ x v + ∂ x d ) + g ′ ( u ξ ) ( ε 1 ⁄ 2 + δ ∂ x 2 v + ∂ x 2 d ) ] ⋅ ∂ x θ d x = F 1 + F 2 + F 3 ,

where u ξ ∈ ( u ¯ a − d , u ¯ a + ε 5 ⁄ 8 v ) , and

(3.14) F 1 = ε 1 ⁄ 2 − δ ∫ R ( g ‴ ( u ξ ) ( ∂ x u ξ ) 2 + g ″ ( u ξ ) ∂ x 2 u ξ ) ( ε 1 ⁄ 2 + δ v + d ) ⋅ ∂ x θ d x ≤ O ( 1 ) ε 1 ⁄ 2 − δ ∫ R ( O ( 1 ) + O ( 1 ) ε 2 ) ( ε 1 ⁄ 2 + δ v + d ) ⋅ ∂ x θ d x ≤ C ε − 3 ∫ R ∣ v ∣ 2 d x + 2 β ε ∫ R ∣ ∂ x θ ∣ 2 d x + C ε 7 γ − 2 δ − 4 ,

F 2 ≤ C ε ∫ R ∂ x u ξ ∂ x v ⋅ ∂ x θ d x + C ε 1 ⁄ 2 − δ ∫ R ∂ x u ξ ∂ x d ⋅ ∂ x θ d x ≤ C ε − 1 ∫ R ∣ ∂ x v ∣ 2 d x + β ε ∫ R ∣ ∂ x θ ∣ 2 d x + C ε − 2 − 2 δ ∫ R ∣ ∂ x d ∣ 2 d x ,

and

F 3 = ε ∫ R g ′ ( u ξ ) ∣ ∂ x θ ∣ 2 d x + ε 1 ⁄ 2 − δ ∫ R g ′ ( u ξ ) ∂ x 2 d ⋅ ∂ x θ d x ,

where

ε 1 ⁄ 2 − δ ∫ R g ′ ( u ξ ) ∂ x 2 d ⋅ ∂ x θ d x ≤ C ε 1 ⁄ 2 − δ ∫ R ∂ x 2 d ⋅ ∂ x θ d x ≤ C ε − 2 δ ∫ R ∣ ∂ x 2 d ∣ 2 d x + β ε ∫ R ∣ ∂ x θ ∣ 2 d x .

We estimate the third term of (3.13) and the detailed calculation process is as follows:

ε − ( 1 ⁄ 2 + δ ) ∫ R ∂ x ( f ( u ¯ a + ε 1 ⁄ 2 + δ v ) − f ( u ¯ a − d ) ) ⋅ ∂ x θ d x ≤ ε − ( 1 ⁄ 2 + δ ) ∫ R ∣ ∂ u 2 f ( u ξ ) ∂ x ( u ξ ) ( ε 1 ⁄ 2 + δ v + d ) ⋅ ∂ x θ ∣ d x + ε − ( 1 ⁄ 2 + δ ) ∫ R ∣ ∂ u f ( u ξ ) ( ε 1 ⁄ 2 + δ ∂ x v + ∂ x d ) ⋅ ∂ x θ ∣ d x ≤ C ε − ( 3 ⁄ 2 + δ ) ∫ R ∣ ( ε 1 ⁄ 2 + δ v + d ) ⋅ ∂ x θ ∣ d x + C ε − ( 1 ⁄ 2 + δ ) ∫ R ∣ ( ε 1 ⁄ 2 + δ ∂ x v + ∂ x d ) ⋅ ∂ x θ ∣ d x ≤ C ε − 1 ∫ R ∣ v ⋅ ∂ x θ ∣ d x + C ε − ( 3 ⁄ 2 + δ ) ∫ R ∣ d ⋅ ∂ x θ ∣ d x + C ∫ R ∣ ∂ x v ⋅ ∂ x θ ∣ d x + C ε − ( 1 ⁄ 2 + δ ) ∫ R ∣ ∂ x d ⋅ ∂ x θ ∣ d x ≤ C ε − 3 ∫ R ∣ v ∣ 2 d x + β ε ∫ R ∣ ∂ x θ ∣ 2 d x + O ( 1 ) ε − 4 − 2 δ ∫ R d 2 d x + β ε ∫ R ∣ ∂ x θ ∣ 2 d x + C ε − 2 − 2 δ ∫ R ∣ ∂ x d ∣ 2 d x + β ε ∫ R ∣ ∂ x θ ∣ 2 d x + C ε − 1 ∫ R ∣ ∂ x v ∣ 2 d x + β ε ∫ R ∣ ∂ x θ ∣ 2 d x ≤ C ε − 3 ∫ R ∣ v ∣ 2 d x + C ε − 1 ∫ R ∣ ∂ x v ∣ 2 d x + 4 β ε ∫ R ∣ ∂ x θ ∣ 2 d x + C ε − 2 − 2 δ ∫ R ∣ ∂ x d ∣ 2 d x + C ε 7 γ − 2 δ − 4 ,

where u ξ ∈ ( u ¯ a − d , u ¯ a + ε 5 ⁄ 8 v ) . For the fourth term of (3.13), we have

ε 1 ⁄ 2 − δ ∫ R ∂ x 2 ( b ( u ¯ a − d ) ∂ x d ) ⋅ θ d x ≤ C ε 1 ⁄ 2 − δ ∫ R ∣ ∂ x b ( u a ) ∂ x d ⋅ ∂ x θ ∣ d x + C ε 1 ⁄ 2 − δ ∫ R ∣ b ( u a ) ∂ x 2 d ⋅ ∂ x θ ∣ d x ≤ C ε − 1 ⁄ 2 − δ ∫ R ∣ ∂ x d ⋅ θ x ∣ d x + C ε − 2 δ ∫ R ∣ ∂ x 2 d ∣ 2 d x + β ε ∫ R ∣ ∂ x θ ∣ 2 d x ≤ C ε − 2 − 2 δ ∫ R ∣ ∂ x d ∣ 2 d x + 2 β ε ∫ R ∣ θ x ∣ 2 d x + C ε − 2 δ ∫ R ∣ ∂ x 2 d ∣ 2 d x .

Collecting all of the aforementioned estimates, we have

1 2 d d t ∫ R ∣ θ ∣ 2 d x + ( C 0 − 10 β ) ε ∫ R ∣ ∂ x θ ∣ 2 d x ≤ C ε 7 γ − 2 δ − 4 + C ε − 2 − 2 δ ∫ R ∣ ∂ x d ∣ 2 d x + C ε − 2 δ ∫ R ∣ ∂ x 2 d ∣ 2 d x + C ε − 3 ∫ R ∣ v ∣ 2 d x + C ε − 1 ∫ R ∣ ∂ x v ∣ 2 d x ≤ C ε 7 γ − 2 δ − 4 + C ε − 1 ∫ R ∣ θ ∣ 2 d x + C ε − 3 ∫ R ∣ ∂ x φ ∣ 2 d x ,

where choosing appropriate β such that ( C 0 − 10 β ) > 0 and using Gronwall’s inequality to obtain (3.11).□

Now, we are in the position to verify that hypothesis (3.5) is correct. It follows from (3.8) and (3.11) that

sup 0 ≤ t ≤ T ‖ ∂ x φ ( ⋅ , t ) ‖ L ∞ ( R ) ≤ 2 sup 0 ≤ t ≤ T ‖ ∂ x φ ( ⋅ , t ) ‖ L 2 ( R ) 1 ⁄ 2 ⋅ sup 0 ≤ t ≤ T ‖ ∂ x 2 φ ( ⋅ , t ) ‖ L 2 ( R ) 1 ⁄ 2 ≤ C ε 7 γ ⁄ 2 − δ − 2 .

Thus, we obtain a unique solution u ε ∈ C 1 ( [ 0 , T ] ; H 2 ( R ) ) and corresponding estimate of the solution for the Cauchy problems (3.2)–(3.3), which yield

(3.15) sup 0 ≤ t ≤ T ‖ u ε ( ⋅ , t ) − u ¯ a ( ⋅ , t ) ‖ L ∞ ( R ) = sup 0 ≤ t ≤ T ‖ ε 1 ⁄ 2 + δ ∂ x φ ( ⋅ , t ) ‖ L ∞ ( R ) ≤ C ε 7 γ ⁄ 2 − 3 ⁄ 2 ,

which complete the proof of Proposition 3.1.

3.5 Proof of Theorem 2.1

Combining the Proposition 3.1 and all of the aforementioned estimates, we finally give the proof of Theorem 1.1. It follows from (3.8) that

sup 0 ≤ t ≤ T ‖ u ε ( ⋅ , t ) − u ¯ a ( ⋅ , t ) ‖ L 2 ( R ) = sup 0 ≤ t ≤ T ‖ ε 1 ⁄ 2 + δ ∂ x φ ( ⋅ , t ) ‖ L 2 ( R ) ≤ C ε 7 γ ⁄ 2 − 1 .

Based on Lemma 2.5 and (3.15), we obtain

sup 0 ≤ t ≤ T ∣ x − s i ( t ) ∣ ≥ ε γ ∣ u ε ( ⋅ , t ) − u 0 ( ⋅ , t ) ∣ ≤ sup 0 ≤ t ≤ T ∣ u ε ( ⋅ , t ) − u ¯ a ( ⋅ , t ) ∣ + sup 0 ≤ t ≤ T ∣ x − s i ( t ) ∣ ≥ ε γ ∣ u ¯ a ( ⋅ , t ) − u 0 ( ⋅ , t ) ∣ ≤ C ε ,

where ∣ x − s i ( t ) ∣ ≥ ε γ , i = 1 , 2 , 6 7 < γ < 1 . Thus, we have completed the proof of Theorem 1.1.

4 Conclusion

To understand the asymptotic relationship between viscous systems and their corresponding inviscid equations is crucial in physics and mathematical analysis, especially in fluid mechanics and numerical computations. This article addresses the inviscid limit problem for the solution of a 1D viscous conservation law. Assuming that the inviscid equation has a piecewise smooth solution with two non-interacting entropy shocks, it is proven that the solutions of the viscous equation uniformly converge to the piecewise smooth inviscid solution away from the shocks, even if the shocks are strong. The mathematical tools in this article such as the multiple-scale asymptotic expansions and energy estimates can also be applied to nonlinear systems to explore the asymptotic behavior of these equations under different conditions.

Acknowledgements

This is the first research work that the author Li Feng participated in during her graduate studies. The author Li Feng thanks her Master’s advisor, Professor Jing Wang, for her encouragement and numerous enlightening discussions. The authors are grateful for the reviewer’s valuable comments that improved the manuscript.

  1. Funding information: The manuscript was supported by National Natural Science Foundation of China (No. 11771297, No. 12261047).

  2. Author contributions: Conceptualization, Jing Wang; methodology, Jing Wang; validation, Jing Wang and Li Feng; formal analysis, Jing Wang and Li Feng; writing-original draft preparation, Li Feng; writing-review and editing, Li Feng; project administration, Jing Wang and Li Feng; funding acquisition, Jing Wang.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-03-26
Revised: 2024-06-11
Accepted: 2024-06-13
Published Online: 2024-11-26

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  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
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  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
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  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0080
Keywords for this article
shock layer; viscous shocks; matched asymptotic expansion; nonlinear stability; energy estimates
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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