The initial-value problem for a Gardner-type equation

Jerry Bona; Hongqiu Chen; Pamela Guerrero; Cristina Haidau; Sanja Pantic

doi:10.1515/ans-2023-0182

Article Open Access

The initial-value problem for a Gardner-type equation

Jerry Bona , Hongqiu Chen , Pamela Guerrero , Cristina Haidau and Sanja Pantic

Published/Copyright: June 5, 2025

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From the journal Advanced Nonlinear Studies Volume 25 Issue 2

Abstract

Discussed here is a regularized version

(0.1) u t + u x + u u x + A u 2 u x − u xxt = 0 ,

of the classical Gardner equation

u t + u x + u u x + A u 2 u x + u xxx = 0 ,

that arises in hydrodynamics and plasma physics. Considered here are well-posedness issues for the initial-value problem for the regularized equation posed on all of R .

Keywords: Gardner equation; bore propagation; long-wave models; unidirectional water-wave models; plasma physics models

2010 AMS Classification: 30D10; 30D15; 30H99; 35B44; 35B65; 35E15; 35Q51; 35Q53

1 Introduction

The Gardner equation

(1.1) u t + u x + u u x + A u 2 u x + u xxx = 0 ,

where A ≠ 0 is a constant and u = u(x, t) is a real-valued function of x ∈ R and t ≥ 0, is named in honor of Clifford Gardner, who first came upon it when analyzing the modified Korteweg–de–Vries (KdV) equation. His equation, in which both quadratic and cubic nonlinearities appear, resulted from a Galilean transformation being applied to the modified Korteweg–de Vries equation. One of the many amazing aspects of the Korteweg–de Vries equation itself falls to Gardner’s analysis, namely the existence of infinitely many conservation laws in the form of polynomials in the solution and its spatial derivatives. This work was published as part of a collection of papers by various combinations of Kruskal, Miura, Gardner, Greene and Su that began the long story of the Korteweg–de Vries equation. (See Miura [1] and references therein for commentary on the early stages of this development; see also Miura [2] for a general derivation of the Gardner equation.)

In later works, the same equation appeared in various guises and often as a model for concrete physical phenomena [3], [4], [5], [6], [7], [8]. This included a model for internal wave propagation and one arising in plasma physics.

It is our purpose here to introduce a so-called regularized version of the Gardner equation, namely

(1.2) u t + u x + u u x + A u 2 u x − u xxt = 0 ,

where A is a constant that remains after u, x and t have all been scaled to present the tidy form (1.2) above.

Equation (1.2) is obtained from the original Gardner equation (1.1) via the observation that the linear dispersion relation relating phase speed c to wavenumber k for (1.2) is

c ( k ) = 1 1 + k 2

while that for the original equation is

c ( k ) = 1 − k 2 .

For small values of k, e.g. large wavelengths, these two expressions differ by a higher-order term. As the Gardner equation presents itself as a model for long wavelength propagation, one can argue that the two models will differ only at higher orders in the small parameters that have been scaled out here. This observation goes back implicitly to Peregrine [9] and more explicitly to the analysis in [10].

In Guerrero’s Ph.D. thesis, it was shown that if the initial data u(x, 0) = φ(x) lies in the L 2 ( R ) -based Sobolev space H s ( R ) for s ≥ 1, then for any A ∈ R , the pure initial-value problem (IVP henceforth) is well-posed globally in time in Hadamard’s classical sense. (Function space notation is briefly reviewed momentarily.) It was also shown that if 1 3 < s < 1 , the problem is globally well-posed if A > 1/6. More recently, Bona, Chen, Hong, Panthee and Scialom [11] showed that if the initial data φ is bore-like together with the condition φ ′ ∈ L 2 ( R ) , then the well-posedness is global without restriction on A. This latter result is particularly telling in the case of certain types of large-amplitude oceanic internal waves (see e.g. [12]).

Remark 1.1.

Initial data is bore-like if it is bounded and has limits at ∞ which are not the same at +∞ and −∞. Such data comes about, for example, when modeling the generation of bores on rivers. In certain situations where a significant river runs into a bay or estuary that experiences high tidal variations in the Spring and Fall, the influx of water raises the water level in the bay/estuary well above that of the incoming river. This has the effect of sending a wave up the river, which resolves into a so-called tidal bore. The relevant initial data that would be fed into a model for two-way propagation of fluid motion would indeed be bore-like in the above sense. References to the mathematical theory of bore propagation include [2], [11], [12], [13], [14], [15], [16], [17].

The present essay extends this previous work by deriving results of local and global well-posedness when the initial data lies in H s ( R ) for 0 ≤ s ≤ 1 3 , in L 1 ( R ) , or L ∞ ( R ) . Here are the new results.

If φ ∈ H s ( R ) ∩ L 4 ( R ) , for 0 ≤ s ≤ 1 3 , then the IVP is locally well-posed. In addition, if A > 1 6 , the local well-posedness is in fact global. That is to say, on any time interval [0, T], u ∈ C ( [ 0 , T ] ; H s ( R ) ∩ L 4 ( R ) ) .

If φ ∈ L 1 ( R ) ∩ L 4 ( R ) , then the IVP is locally well-posed. Moreover, it is globally well posed if A > 1 6 .

Finally, if φ ∈ L 1 ∩ L ∞ , then the IVP is globally well-posed for any A ∈ R , which is to say, u ∈ C ( [ 0 , ∞ ) ; L 1 ( R ) ∩ L ∞ ( R ) ) .

These improved results stem from the observation that, in addition to a Hamiltonian that can provide a priori bounds, the Gardner equation possesses a second conserved quantity that can be used independently of the Hamiltonian.

The plan of the paper is straightforward. Section 2 deals briefly with notation and some elementary results that find repeated use later. The heart of the discussion is in Section 3 where the new well-posedness results are derived.

2 Notation and preliminaries

2.1 Notation

For 1 ≤ p < ∞, L p = L p ( R ) is the standard pth-power integrable Lebesgue space with its norm denoted by |f|_p. The usual modification is in force when p = +∞. The norm on the L 2 ( R ) -based Sobolev spaces H s = H s ( R ) for s ≥ 0 is

(2.3) ‖ f ‖ s = ∫ ( 1 + ξ 2 ) s | f ̂ ( ξ ) | 2 d ξ 1 2 ,

where the Fourier transform f ̂ of an L 2 ( R ) -function is

f ̂ ( ξ ) = 1 2 π ∫ f ( x ) e − i x ξ d x .

When s = 0, H 0 ( R ) = L 2 ( R ) and ‖f‖₀ = |f|₂ is simply replaced with the unembellished notation ‖f‖. Throughout the paper, the unadorned integral ∫ means ∫ R . If the space over which functions from some class is not delineated, it is always assumed to be the whole real line R . Thus, L 2 = L 2 ( R ) etc.

If X is a Banach space and T > 0 is finite, the collection

X T = C ( [ 0 , T ] ; X ) = { f : [ 0 , T ] ↦ X continuously }

is a Banach space when equipped with the norm

‖ f ‖ X T = sup 0 ≤ t ≤ T ‖ f ( t ) ‖ X = max 0 ≤ t ≤ T ‖ f ( t ) ‖ X .

The space C([0, ∞); X) connotes those mappings from the non-negative real axis [0, ∞) → X such that they lie in C([0, T]; X) for all T > 0. It is naturally a Fréchet space, but its topology will not be needed.

2.2 Preliminaries

The following elementary relations will be used repeatedly.

Convolution equality:

f ∗ g ̂ ( ξ ) = 2 π f ̂ ( ξ ) g ̂ ( ξ ) .

Parseval-Plancherel identy:

‖ f ‖ = ‖ f ̂ ‖ .

Hausdorff–Young inequality: If f ∈ L _q where 1 ≤ q ≤ 2, then

| f ̂ | p ≤ | f | q

where p = q q − 1 .

Young’s convolution inequality: If 1 ≤ p, q, r ≤ ∞ satisfy

1 p + 1 q = 1 r + 1

and f ∈ L _p, g ∈ L _q, then f∗g ∈ L _r and

| f ∗ g | r ≤ | f | p | g | q .

The next lemma features elementary technical results about the mapping properties of convolution with the kernel

(2.4) K ( x ) = 1 2 sgn ( x ) e − | x | ,

whose Fourier transform is

(2.5) K ̂ ( ξ ) = − i ξ 2 π ( 1 + ξ 2 ) .

This convolution kernel arises from the corespondence

(2.6) F ↦ − ∂ x I − ∂ x 2 − 1 F = ∫ K ∗ F = K ( F )

that plays a significant role in our developments.

Lemma 2.1.

The correspondence (2.6), when applied to u ∈ X 4 s = L 4 ( R ) ∩ H s ( R ) with 0 ≤ s ≤ 1 3 , has the following mapping properties:

(2.7) ‖ K ∗ u ( ⋅ , τ ) ‖ s ≤ ‖ u ( ⋅ , τ ) ‖ , ‖ K ∗ u 2 ( ⋅ , τ ) ‖ s ≤ | u ( ⋅ , τ ) | 4 2 ,

and

(2.8) ‖ K ∗ u 3 ( ⋅ , τ ) ‖ s ≤ 2 | u ( ⋅ , τ ) | 4 3 .

Proof.

These inequalities follow from the estimates:

(2.9) ‖ K ∗ u ( ⋅ , τ ) ‖ s 2 = ∫ ( 1 + ξ 2 ) s | K ∗ u ̂ ( ξ , τ ) | 2 d ξ = ∫ ( 1 + ξ 2 ) s ξ 2 ( 1 + ξ 2 ) 2 | u ̂ ( ξ , τ ) | 2 d ξ ≤ ∫ | u ̂ ( ξ , τ ) | 2 d ξ = ‖ u ( ⋅ , τ ) ‖ 2 ;

(2.10) ‖ K ∗ u 2 ( ⋅ , τ ) ‖ s 2 = ∫ ( 1 + ξ 2 ) s | K ∗ u 2 ̂ ( ξ , τ ) | 2 d ξ = ∫ ( 1 + ξ 2 ) s ξ 2 ( 1 + ξ 2 ) 2 | u 2 ̂ ( ξ , τ ) | 2 d ξ , ≤ ∫ | u 2 ̂ ( ξ , τ ) | 2 d ξ = ‖ u 2 ( ⋅ , τ ) ‖ 2 = | u ( ⋅ , τ ) | 4 4 .

For the cubic term, write

‖ K ∗ u 3 ( ⋅ , τ ) ‖ s 2 = ∫ ( 1 + ξ 2 ) s | K ∗ u 3 ̂ ( ξ , τ ) | 2 d ξ = ∫ ( 1 + ξ 2 ) s ξ 2 ( 1 + ξ 2 ) 2 | u 3 ̂ ( ξ , τ ) | 2 d ξ .

First apply the Cauchy–Schwarz inequality and then the Hausdorff–Young inequality to obtain

(2.11) ‖ K ∗ u 3 ( ⋅ , τ ) ‖ s 2 ≤ ∫ ( 1 + ξ 2 ) 2 s ξ 4 ( 1 + ξ 2 ) 4 d ξ 1 2 ∫ | u 3 ̂ ( ξ , τ ) | 4 d ξ 1 2 ≤ ∫ ( 1 + ξ 2 ) 2 s ξ 4 ( 1 + ξ 2 ) 4 d ξ 1 2 | u 3 ( ⋅ , τ ) | 4 3 2 ≤ 2 | u ( ⋅ , τ ) | 4 6 .

□

3 Well-posedness

Following a standard analysis, the IVP

(3.12) u t + u x + u u x + A u 2 u x − u xxt = 0 , x ∈ R , t ≥ 0 , u ( x , 0 ) = φ ( x ) ,

is converted to an equivalent integral equation by first inverting I − ∂ x 2 , viz.

(3.13) u t ( x , t ) = − 1 2 ∫ e − | x − y | ∂ y u ( y , t ) + 1 2 u 2 ( y , t ) + 1 3 A u 3 ( y , t ) d y ,

and then integrating in time over [0, t] to reach

(3.14) u ( x , t ) = φ ( x ) − 1 2 ∫ 0 t ∫ e − | x − y | ∂ y u ( y , t ) + 1 2 u 2 ( y , t ) + 1 3 A u 3 ( y , t ) d y d t .

After an integration by parts, there appears the integral equation

(3.15) u ( x , t ) = φ ( x ) + ∫ 0 t ∫ K ( x − y ) u ( y , t ) + 1 2 u 2 ( y , t ) + 1 3 A u 3 ( y , t ) d y d t ≕ A u ( x , t ) ,

where K is the exponential kernel introduced in (2.4).

Fixed points of A correspond to weak, strong or classical solutions of (3.12), depending upon the function class in which they lie. For example, if a bounded and continuous function u satisfies (3.15) for t ∈ [0, T], then it is a classical solution of (3.12) on R × [ 0 , T ] .

Introduce the linear space

(3.16) X 4 s = H s ∩ L 4 = H s ( R ) ∩ L 4 ( R ) .

It is a Banach space when equipped with the norm

‖ f ‖ s , 4 = max { ‖ f ‖ s , | f | 4 } .

Lemma 3.1.

If φ ∈ X 4 s ( R ) = H s ( R ) ∩ L 4 ( R ) , where 0 ≤ s ≤ 1 3 , then the initial-value problem (3.12) is locally well-posed. That is to say, there is a positive number T which only depends on the norms ‖φ‖_s and |φ|₄ such that (3.12), or (3.15), has a unique solution u ∈ X T = C [ 0 , T ] ; X 4 s ; furthermore, the solution mapping which takes initial data φ ∈ X 4 s to the corresponding solution u ∈ C [ 0 , T ] ; X 4 s is locally Lipschitz continuous.

Proof.

The contraction-mapping theorem is applied to the operator A defined in (3.15). To begin, suppose that u is arbitrary in X _T with initial value φ. Then for any t ∈ [0, T],

(3.17) ‖ A u ( , t ) ‖ s ≤ ‖ φ ‖ s + ∫ 0 t ‖ K ∗ u ( ξ , τ ) ‖ s + 1 2 ‖ K ∗ u 2 ( ⋅ , τ ) ‖ s + 1 3 | A | ‖ K ∗ u 3 ( ⋅ , τ ) ‖ s d τ .

The terms on the right-hand side are easily estimated via Lemma 2.1 as follows:

(3.18) ‖ A u ( ⋅ , t ) ‖ s ≤ ‖ φ ‖ s + ∫ 0 t ‖ u ( ⋅ , τ ) ‖ + 1 2 | u ( ⋅ , τ ) | 4 2 + 2 3 | A | u ( ⋅ , τ ) | 4 3 d τ .

That is to say, the operator A maps the space X _T to C([0, T]; H ^s). It remains to verify that it also maps the space X _T to C([0, T]; L ₄). This follows by applying Young’s convolution inequality, viz.

(3.19) | A u ( ⋅ , t ) | 4 ≤ | φ | 4 + ∫ 0 t | K ∗ u ( ⋅ , τ ) | 4 + 1 2 | K ∗ u 2 ( ⋅ , τ ) | 4 + 1 3 | A | | K ∗ u 3 ( ⋅ , τ ) | 4 d τ ≤ | φ | 4 + ∫ 0 t | K | 4 3 ‖ u ( ⋅ , τ ) ‖ + 1 2 | K | 4 3 ‖ u 2 ( ⋅ , τ ) ‖ + 1 3 | A | ‖ K ‖ | u 3 ( ⋅ , τ ) | 4 3 d τ ≤ | φ | 4 + ∫ 0 t ‖ u ( ⋅ , τ ) ‖ + 1 2 | u ( ⋅ , τ ) | 4 2 + 1 3 | A ‖ u ( ⋅ , τ ) | 4 3 d τ .

Thus it is established that the operator A maps the space X _T to itself for any T > 0.

Attention is turned now to showing that when T > 0 is well chosen, A has a fixed point in X _T. To this end, fix initial data φ ∈ X 4 s . Let

r = 2 ‖ φ ‖ s , 4

and choose

T = 1 2 ( 1 + r + 2 | A | r 2 ) .

We claim that the operator A is contractive on the closed ball

B r = { u ∈ X T : ‖ u ‖ X T ≤ r } .

Indeed, for any u , v ∈ B r , (3.18) and (3.19) imply that

(3.20) ‖ A u ‖ X T ≤ ‖ φ ‖ X s , 4 + T ‖ u ‖ X T + 1 2 ‖ u ‖ X T 2 + 2 3 | A | ‖ u ‖ X T 3 ≤ 1 2 r + T 1 + 1 2 r + 2 3 | A | r 2 r ≤ r ,

and

(3.21) ‖ A u ( ⋅ , t ) − A v ( ⋅ , t ) ‖ s ≤ ∫ 0 t ‖ K ∗ ( u ( ⋅ , τ ) − v ( ⋅ , τ ) ) ‖ s + 1 2 ‖ K ∗ ( u 2 ( ⋅ , τ ) − v 2 ( ⋅ , τ ) ) ‖ s + 1 3 | A | ‖ K ∗ ( u 3 ( ⋅ , τ ) − v 3 ( ⋅ , τ ) ) ‖ s d τ .

The estimates leading to (2.7) and (2.8) imply that

(3.22) ‖ A u ( ⋅ , t ) − A v ( ⋅ , t ) ‖ s ≤ ∫ 0 t ‖ u ( ⋅ , τ ) − v ( ⋅ , τ ) ‖ + 1 2 | u 2 ( ⋅ , τ ) − v 2 ( ⋅ , τ ) | 2 + 2 3 | A | | u 3 ( ⋅ , τ ) − v 3 ( ⋅ , τ ) | 4 3 d τ .

The Hölder inequality gives

‖ u 2 ( ⋅ , τ ) − v 2 ( ⋅ , τ ) ‖ = ‖ u ( ⋅ , τ ) + v ( ⋅ , τ ) u ( ⋅ , τ ) − v ( ⋅ , τ ) ‖ ≤ | u ( ⋅ , τ ) + v ( ⋅ , τ ) | 4 u ( ⋅ , τ ) − v ( ⋅ , τ ) 4 ≤ | u ( ⋅ , τ ) | 4 + | v ( ⋅ , τ ) | 4 u ( ⋅ , τ ) − v ( ⋅ , τ ) 4 ≤ ‖ u ‖ X T + ‖ v ‖ X T ‖ u − v ‖ X T ≤ 2 r ‖ u − v ‖ X T ,

and in the same fashion,

| u 3 ( ⋅ , τ ) − v 3 ( ⋅ , τ ) | 4 3 = | u 2 ( ⋅ , τ ) + u ( ⋅ , τ ) v ( ⋅ , τ ) + v 2 ( ⋅ , τ ) u ( ⋅ , τ ) − v ( ⋅ , τ ) | 4 3 ≤ u 2 ( ⋅ , τ ) + u ( ⋅ , τ ) v ( ⋅ , τ ) + v 2 ( ⋅ , τ ) 2 u ( ⋅ , τ ) − v ( ⋅ , τ ) 4 ≤ | u ( ⋅ , τ ) | 4 2 + | u ( ⋅ , τ ) | 4 | v ( ⋅ , τ ) | 4 + | v ( ⋅ , τ ) | 4 2 u ( ⋅ , τ ) − v ( ⋅ , τ ) 4 ≤ ‖ u ‖ X T 2 + ‖ u ‖ X T ‖ v ‖ X T + ‖ v ‖ X T 2 ‖ u − v ‖ X T ≤ 3 r 2 ‖ u − v ‖ X T .

Insert the last two inequalities into (3.22) to come to

(3.23) ‖ A u ( ⋅ , t ) − A v ( ⋅ , t ) ‖ s ≤ ∫ 0 t ‖ K ∗ ( u ( ⋅ , τ ) − v ( ⋅ , τ ) ) ‖ s + 1 2 ‖ K ∗ ( u 2 ( ⋅ , τ ) − v 2 ( ⋅ , τ ) ) ‖ s + 2 3 | A | ‖ K ∗ ( u 3 ( ⋅ , τ ) − v 3 ( ⋅ , τ ) ) ‖ s d τ ≤ T 1 + r + 2 | A | r 2 ‖ u − v ‖ X T = 1 2 ‖ u − v ‖ X T .

Young’s convolution inequality and another application of Hölder’s inequality yield

(3.24) | A u ( ⋅ , t ) − A v ( ⋅ , t ) | 4 ≤ ∫ 0 t | u ( ⋅ , τ ) − v ( ⋅ , τ ) | 4 + 1 2 | u 2 ( ⋅ , τ ) − v 2 ( ⋅ , τ ) | 2 + 1 3 | A ‖ u 3 ( ⋅ , τ ) − v 3 ( ⋅ , τ ) | 4 3 d τ ≤ T 1 + r + | A | r 2 ‖ u − v ‖ C ( [ 0 , T ] : L 4 ) ≤ 1 2 ‖ u − v ‖ X T .

Inequality (3.23) together with (3.24) indicate that

(3.25) ‖ A u − A v ‖ X T ≤ 1 2 ‖ u − v ‖ X T .

The estimates (3.20) and (3.25) show that A is, indeed, a contractive mapping on the closed ball B r . Therefore, it has a unique fixed point u ∈ B r . It remains to show the continuity of this fixed point with respect to variations in the initial data.

Suppose that u, v are two solutions of the integral equation (3.15) with initial data φ and ψ, respectively, both lying in B r . Introduce the notation A subscripted with the initial data φ to distinguish between the operators. Thus,

u = A φ ( u ) a n d v = A ψ ( v ) .

Write u − v in the form

u − v = A φ ( u ) − A φ ( v ) + A φ ( v ) − A ψ ( v )

so that

‖ u − v ‖ X T ≤ 1 2 ‖ u − v ‖ X T + ‖ φ − ψ ‖ X s , 4 ,

showing the data to solution map to be locally Lipschitz continuous.□

Lemma 3.2.

Still assuming that the initial data φ lies in X 4 s = H s ∩ L 4 , the difference u − φ is smoother in x than either φ or u. In fact,

(3.26) u − φ ∈ C [ 0 , T ] : H 3 4 − ϵ ( R )

for any ϵ > 0.

Proof.

As u is a solution of the integral equation (3.15),

(3.27) u − φ = ∫ 0 t K ∗ u + 1 2 u 2 + 1 3 A u 3 d t .

Since u ∈ C [ 0 , T ] ; X 4 s ⊂ C ( [ 0 , T ] ; L 2 ∩ L 4 ) for 0 ≤ s ≤ 1 3 , u ² ∈ C([0,T]; L ₂), so K ∗ u + 1 2 u 2 ∈ C ( [ 0 , T ] ; H 1 ) ⊂ C ( [ 0 , T ] ; H 3 4 − ϵ ) .

Let σ < 3 4 be arbitrary. By the Cauchy–Schwarz inequality,

(3.28) ‖ K ∗ u 3 ( ⋅ , t ) ‖ σ 2 = ∫ ( 1 + ξ 2 ) σ ξ 2 ( 1 + ξ 2 ) 2 u 3 ̂ ( ξ , t ) 2 d ξ ≤ ∫ ( 1 + ξ 2 ) σ ξ 2 ( 1 + ξ 2 ) 2 2 d ξ 1 2 ∫ | u 3 ̂ ( ξ , t ) 4 d ξ 1 2 = c 2 | u 3 ̂ ( ⋅ , t ) | 4 2 ,

where

c = ∫ ( 1 + ξ 2 ) σ ξ 2 ( 1 + ξ 2 ) 2 2 d ξ 1 4 < ∞

since σ < 3 4 . Young’s convolution inequality then implies that

(3.29) ‖ K ∗ u 3 ( ⋅ , t ) ‖ σ 2 ≤ c 2 | u 3 ( ⋅ , t ) | 4 3 2 = c 2 | u ( ⋅ , t ) | 4 6 ,

which is to say that

‖ K ∗ u 3 ( ⋅ , t ) ‖ σ ≤ c | u ( ⋅ , t ) | 4 3

and thus

(3.30) K ∗ u ( ⋅ , t ) + 1 2 u ( ⋅ , t ) 2 + 1 3 A u 3 ( ⋅ , t ) ∈ H σ ( R ) .

The formula (3.27) then allows the conclusion

u − φ ∈ C ( [ 0 , T ] ; H σ ) .

□

Theorem 3.3.

Assume that the constant A > 1 6 in the regularized Gardner equation, and the initial data φ ∈ X 4 s . Then the IVP is globally well posed, which is to say u ∈ C ( [ 0 , ∞ ) ; X 4 s ) .

Proof.

It is straightforward to verify that

d d t ∫ u 2 ( x , t ) + 1 3 u 3 ( x , t ) + 1 6 A u 4 ( x , t ) d x = 0 .

Formally, one just passes the time-derivative through the integral and uses the differential equation. One can justify this by regularizing the initial data, making the calculations securely with the associated smooth solutions and when we arrive at

(3.31) ∫ u 2 ( x , t ) + 1 3 u 3 ( x , t ) + 1 6 A u 4 ( x , t ) d x = ∫ φ 2 ( x ) + 1 3 φ 3 ( x ) + 1 6 A φ 4 ( x ) d x ,

take the limit as the regularization goes away. As solutions depend continuously on the initial data, the relation (3.31) is seen to hold even for data in X 4 s . The same conclusion can be found by instead regularizing the solutions as in [17]. Since A > 1 6 ,

1 3 | u | 3 ≤ 1 6 A ϵ u 2 + ϵ 6 A u 4

for any ϵ > 0. This in turn implies that

(3.32) ∫ u 2 ( x , t ) + 1 3 u 3 ( x , t ) + 1 6 A u 4 ( x , t ) d x ≥ ∫ 1 − 1 6 A ϵ u 2 ( x , t ) + 1 6 A ( 1 − ϵ ) u 4 ( x , t ) d x .

Choosing ϵ = 1 6 A reveals that

(3.33) 1 − 1 6 A ∫ u 2 ( x , t ) + 1 6 A u 4 ( x , t ) d x ≤ c 0 1 − 1 6 A ,

where

c 0 = 1 − 1 6 A − 1 ∫ φ 2 ( x ) + 1 2 φ 3 ( x ) + 1 6 A φ 4 ( x ) d x

is a positive constant dependent only on the initial data. It follows readily that

(3.34) ‖ u ( ⋅ , t ) ‖ ≤ c 0 as well as | u ( ⋅ , t ) | 4 ≤ c 2 = 6 A c 0 1 4 .

As this holds for any t > 0, the local theory can be iterated indefinitely to yield global well posedness.□

Corollary 3.4.

If φ ∈ L 1 ( R ) ∩ L 4 ( R ) , then the initial-value problem (3.12) is locally well-posed. The local well-posedness extends to global if A > 1 6 .

Proof.

If φ ∈ L ₁ ∩ L ₄, then φ ∈ L ₂ ∩ L ₄. By Lemma 3.1, the initial value problem (3.12) is locally well-posed and there are values T = T(‖φ‖, |φ|₄) > 0 such that the solution u lies in X _T = C([0, T]; L ₂ ∩ L ₄).

For each t ∈ [0, T], u ( ⋅ , t ) ∈ L j ( R ) for j = 2, 3, hence u ²(⋅, t) and u ³(⋅, t) are both L 1 ( R ) -functions. The operator K is a bounded linear operator mapping L ₁ to L ₁ with | K f | 1 ≤ | K | 1 | f | 1 = | f | 1 . Hence, K generates a continuous semigroup e K t on L 1 ( R ) . Apply the Duhamel formula to equation (3.13),

u t = K ∗ u + 1 2 u 2 + 1 3 A u 3 ,

repeated for convenience, to deduce that

u ( ⋅ , t ) = e K t φ ( ⋅ ) + ∫ 0 t e K ( t − τ ) 1 2 u 2 ( ⋅ , τ ) + A 3 u 3 ( ⋅ , τ ) d τ .

Of course, e K t φ ( ⋅ ) ∈ C ( [ 0 , ∞ ) ; L 1 ( R ) ) since φ ∈ L 1 ( R ) . Elementary inequalities together with Lemma 2.1 imply that

(3.35) ∫ 0 t e K ( t − τ ) 1 2 u 2 ( ⋅ , τ ) + A 3 u 3 ( ⋅ , τ ) d τ 1 ≤ ∫ 0 t e t − τ 1 2 | u 2 ( ⋅ , τ ) | 1 + | A | 3 | u 3 ( ⋅ , τ ) | 1 d τ ≤ ( e t − 1 ) 1 2 ‖ u ‖ X T 2 + | A | 3 ‖ u ‖ X T 3

since

K ∗ u 3 ( ⋅ , t ) 1 ≤ | K | 1 u ( ⋅ , t ) 3 3 ≤ ‖ u ( ⋅ , t ) ‖ u ( ⋅ , t ) 4 2 ≤ ‖ u ‖ X T 3 .

Thus the local well posedness in C ( [ 0 , T ] ; L 1 ( R ) ∩ L 4 ( R ) ) follows.

If A > 1/6, Theorem 3.3 has it that ‖ u ‖ X T < ∞ for any T > 0. Hence the preceding estimates apply for any T > 0 and the corollary is established.□

Corollary 3.5.

If φ ∈ L ∞ ( R ) ∩ L 2 ( R ) , then the IVP is globally well-posed for any A ≠ 0. That is to say, on any time interval [0, T], |u(⋅, t)|_∞ and |u(⋅, t)|₂ are bounded for t ∈ [0, T].

Proof.

As in the last corollary, (3.12) has a unique local solution u lying in the space C ( [ 0 , T ) ; L 2 ( R ) ∩ L 4 ( R ) ) for some T = T(‖φ‖, |φ|₄) > 0. Since for any t ∈ [0, T],

K ∗ u ( ⋅ , t ) + 1 2 u 2 ( ⋅ , t ) + 1 3 A u 3 ( ⋅ , t ) ∞ ≤ ‖ K ‖ ‖ u ( ⋅ , t ) ‖ + 1 2 ‖ K ‖ | u ( ⋅ , t ) | 4 2 + 1 3 | A | | K | 4 | u 3 ( ⋅ , t ) | 4 3 ≤ ‖ u ( ⋅ , t ) ‖ + 1 2 | u ( ⋅ , t ) | 4 2 + 1 3 | A | | u ( ⋅ , t ) | 4 3 ,

it follows that for t ∈ [0, T],

u ( ⋅ , t ) = φ + ∫ 0 t K ∗ u ( ⋅ , t ) + 1 2 u 2 ( ⋅ , t ) + 1 3 A u 3 ( ⋅ , t ) d t ∈ L ∞ ( R ) .

Once u ∈ C([0, T]; L ₂ ∩ L _∞), it follows that u + 1 2 u 2 + 1 3 A u 3 ∈ L 2 ( R ) , and so

u − φ ∈ C ( [ 0 , T ] ; H 1 ( R ) ) .

Global well-posedness will follow as soon as a priori bounds are in place. To this end, let v = u − φ. The initial-value problem (3.12) in terms of v becomes

(3.36) v t + v x + v v x + A v 2 v x − v xxt + ( φ v ) x + A φ 2 v + φ v 2 x + φ + 1 2 φ 2 + 1 3 A φ 3 x = 0 , v ( x , 0 ) = 0 , x ∈ R , t ≥ 0 .

Multiply the last equation by v and integrate with respect to x over R . After suitable integrations by parts, there obtains

(3.37) 1 2 d d t ∫ v 2 ( x , t ) + v x 2 ( x , t ) d t = ∫ φ + A φ 2 v ( x , t ) v x ( x , t ) + A φ v 2 ( x , t ) v x ( x , t ) + φ + 1 2 φ 2 + A 3 φ 3 v x ( x , t ) d x ≤ φ + A φ 2 ∞ ‖ v ( ⋅ , t ) ‖ ‖ v x ( ⋅ , t ) ‖ + | A ‖ φ | ∞ | v ( ⋅ , t ) | 4 2 ‖ v x ( ⋅ , t ) ‖ + 1 + 1 2 φ + A 3 φ 2 ∞ ‖ φ ‖ ‖ | v x ( ⋅ , t ) ‖ .

Observe that

| A ‖ φ | ∞ | v ( ⋅ , t ) | 4 2 ‖ v x ( ⋅ , t ) ‖ ≤ 1 4 A 2 | v ( ⋅ , t ) | 4 4 + | φ | ∞ 2 ‖ v x ( ⋅ , t ) ‖ 2 ,

and notice also that

‖ v ( ⋅ , t ) ‖ ‖ v x ( ⋅ , t ) ‖ ≤ 1 2 ‖ v ( ⋅ , t ) ‖ 2 + 1 2 ‖ v x ( ⋅ , t ) ‖ 2 = 1 2 ‖ v ( ⋅ , t ) ‖ 1 2 .

One concludes immediately that

(3.38) 1 2 d d t ‖ v ( ⋅ , t ) ‖ 1 2 ≤ 1 2 φ + A φ 2 ∞ ‖ v ( ⋅ , t ) ‖ 1 2 + | φ | ∞ 2 ‖ v x ( ⋅ , t ) ‖ 2 + 1 4 A 2 v ( ⋅ , t ) 4 4 + | 1 + 1 2 φ + 1 3 A φ 2 | ∞ ‖ φ ‖ ‖ v ( ⋅ , t ) ‖ 1 ≤ | 1 + 1 2 φ + 1 3 A φ 2 | ∞ ‖ φ ‖ ‖ v ( ⋅ , t ) ‖ 1 + 1 2 ( 1 + | A ‖ φ | ∞ + 2 | φ | ∞ ) | φ | ∞ ‖ v ( ⋅ , t ) ‖ 1 2 + 1 4 A 2 | v ( ⋅ , t ) | 4 4 .

It remains to deal with the term | v ( ⋅ , t ) | 4 4 . For the purpose of obtaining helpful bounds on this term, recall that if (3.12) has a solution u ∈ C([0, T]; L ₂ ∩ L ₄), then the identity

(3.39) ∫ u 2 ( x , t ) + 1 3 u 3 ( x , t ) + 1 6 A u 4 ( x , t ) d x = ∫ φ 2 ( x ) + 1 3 φ 3 ( x ) + 1 6 A φ 4 ( x ) d x

holds (see (3.31)). Replace u = u(x, t) with v + φ = v(x, t) + φ(x) and multiply both sides by 6A to obtain

(3.40) A 2 ∫ v 4 + 4 φ v 3 + 6 φ 2 v 2 + 2 A v 3 d x = − A ∫ 6 v 2 + 6 φ v 2 + 4 A φ 3 v + 6 φ 2 v + 12 φ v d x ≤ 6 | A | | 1 + φ | ∞ ‖ v ‖ 2 + 2 | A | 6 + 3 φ + 2 A φ 2 ∞ ‖ φ ‖ ‖ v ‖ .

Remark that

v 4 + 4 φ v 3 + 6 φ 2 v 2 ≥ 1 3 v 4 and 2 A v 3 ≥ − 1 6 v 4 − 6 A 2 v 2 .

Insert these relations into (3.40) to come to the inequality

1 6 A 2 | v ( ⋅ , t ) | 4 4 − 1 6 ‖ v ( ⋅ , t ) ‖ 2 ≤ 6 | A | | 1 + φ | ∞ ‖ v ‖ 2 + 2 | A | 6 + 3 φ + 2 A φ 2 ∞ ‖ φ ‖ ‖ v ‖ .

Rewriting the latter inequality as

(3.41) A 2 | v ( ⋅ , t ) | 4 4 ≤ 36 | A | | 1 + φ | ∞ + 1 ‖ v ‖ 2 + 12 | A | 6 + 3 φ + 2 A φ 2 ∞ ‖ φ ‖ ‖ v ‖

and inserting it into (3.38) leads to the differential inequality

(3.42) 1 2 d d t ‖ v ( ⋅ , t ) ‖ 1 2 ≤ 18 | A | + 1 1 + 1 2 φ + 1 3 A φ 2 ∞ ‖ φ ‖ ‖ v ( ⋅ , t ) ‖ 1 + 1 2 ( 1 + | φ | ∞ + | A ‖ φ | ∞ ) | φ | ∞ ‖ v ( ⋅ , t ) ‖ 1 2 + 9 | A | | 1 + φ | ∞ + 1 4 ‖ v ( ⋅ , t ) ‖ 2 ≤ θ 0 ‖ v ( ⋅ , t ) ‖ 1 + θ 1 ‖ v ( ⋅ , t ) ‖ 1 2 ,

where

(3.43) θ 0 = 1 + 18 | A | 1 + 1 2 | φ | ∞ + 1 3 | A ‖ φ | ∞ 2 ‖ φ ‖ and θ 1 = 1 4 + 9 | A | + 1 2 + 9 | A | | φ | ∞ + 1 2 ( 1 + | A | ) | φ | ∞ 2 .

A Gronwall-type argument thus implies that

‖ v ( ⋅ , t ) ‖ 1 ≤ θ 0 ∫ 0 t e θ 1 ( t − τ ) d τ = θ 0 θ 1 e θ 1 t − 1 ,

which provides a time-dependent, but finite bound on the H 1 ( R ) -norm of v for any time t. Hence v, and therefore u lies in C([0, ∞); L ₂ ∩ L _∞).□

Remark 3.6.

In the case A = 0, the regularized Gardner equation reduces to the BBM-equation. In Chen [18], it was shown that if the initial data φ ∈ L ₂, then the BBM-equation is globally well-posed and u − φ − t K ∗ φ + 1 2 φ 2 ∈ C ( [ 0 , ∞ ) ; H 1 ) . It is straightforward to see that this result can be extended, showing that if φ ∈ L ₂ ∩ L _∞, then u ∈ C([0, ∞); L ₂ ∩ L _∞).

4 Remark on more general BBM-type equations

The foregoing analysis can be extended to a more general class of equations without much modification. One direction for extension is IVPs of the form

(4.44) u t + u x + u u x + f ( u ) x − u xxt = 0 , x ∈ R , t ≥ 0 , u ( x , 0 ) = φ ( x )

where f(x) is a polynomial function for which f(0) = f′(0) = f″(0) = 0. If φ ∈ H s ( R ) for s ≥ 1, then this IVP is globally well-posed.

Furthermore, if the highest order monomial of f(x) is odd, say of order 2p + 1 where p ≥ 1, and there are two positive numbers λ ₁ and λ ₂ such that

1 2 x 2 + 1 6 x 3 + F ( x ) ≥ λ 1 x 2 + λ 2 x 2 p + 2 ,

where F ( x ) = ∫ 0 x f ( x ) d x , then slight generalizations of the foregoing analysis reveal that the problem is globally well-posed for φ ∈ H s ( R ) for any s in the range 1 2 − 1 2 ( 2 p + 1 ) < s < 1 . Moreover, the solution u has the temporal bound

‖ u ( ⋅ , t ) ‖ s ≤ ‖ φ ‖ s + c 1 t

where c ₁ is a constant and

u − φ ∈ C ( [ 0 , ∞ ) ; H 1 ( R ) ) .

If φ ∈ H s ( R ) ∩ L 2 p + 2 ( R ) for 0 ≤ s ≤ 1 2 − 1 2 ( 2 p + 1 ) , then it is globally well-posed with

‖ u ( ⋅ , t ) ‖ s ≤ ‖ φ ‖ s + c 2 t ,

where c ₂ is another constant and

u − φ ∈ C ( [ 0 , ∞ ) ; H σ − )

where σ = 1 2 + 1 2 ( p + 1 ) .

Corresponding author: Jerry Bona, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street. M/C 249, Chicago, IL 60607, USA, E-mail: jbona@uic.edu

Dedicated to Robert Fefferman, distinguished former colleague and long-time friend of Bona.

Acknowledgments

Much of the writing was done while JB and HC were in residence at UNIST (Ulsan National University of Science and Technology). They are grateful for hospitality and excellent working conditions. The L 1 and L ∞ results were derived during conversations between HC, Junsik Bae and Min Gie while she was visiting KAIST (Korea Advanced Institute of Science and Technology).

Research ethics: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Research funding: JB thanks UIC (University of Illinois at Chicago) for research support during this collaboration. PG was partly supported by a Student Success and Innovation Ph.D. Fellowship at the University of Memphis during the development of this project.
Data availability: Not applicable.

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Received: 2025-02-18

Accepted: 2025-04-04

Published Online: 2025-06-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2023-0182

Keywords for this article

Gardner equation; bore propagation; long-wave models; unidirectional water-wave models; plasma physics models

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