Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed

1 Introduction

The compressible fluid models of Korteweg type were introduced by Korteweg [23] and deduced rigorously in Dunn and Serrin [9]. It governs the motions of the compressible isothermal viscous capillary fluids and is formulated as

Here, ( x , t ) ∈ R d × R + , the unknown functions ρ ˜ ( x , t ) and u ( x , t ) represent fluid density and fluid velocity, respectively. P = P ( ρ ˜ ) is the pressure given by a smooth function depending on ρ ˜ , and u ⊗ u denotes the tensor product of two vectors. The viscosity constants μ and μ ′ satisfy μ > 0 and 2 μ + μ ′ > 0 , and κ > 0 is the capillary coefficient.

One of the most important features of system (1.1) is that the pressure is in general non-monotone. For instance, P ′ ( ρ ¯ ) < 0 ( ρ ¯ is the background density) will lead to linear instability for some value of ρ ¯ . Thus, almost all of the mathematical results (in the whole space) only deal with the condition that P ′ ( ρ ¯ ) > 0 , except for local-in-time results. In fact, there have been a lot of mathematical studies on system (1.1), which include the weak solutions in [2,12], the existence in Besov spaces in [4,21], the local existence of strong solutions in [24], the global existence of strong solutions in [37], the maximal L p - L q regularity in [31], the global existence of smooth solution in [13,14], and L 2 -decay rate of the smooth solution in [10,38]. With regard to the nonisothermal case, we refer to [16,17] for the global existence and the decay rate of the smooth solution with small initial energy, respectively, and [18] for the vanishing capillarity limit. When the sound speed c = P ′ ( ρ ¯ ) > 0 , Jiang and Wu [19] verified the pointwise space-time behavior of the classical solution to the Cauchy problem of (1.1) as

which means that the pointwise decription contains two diffusion waves, one is D-wave (zero sound speed) and the other is H-wave (the sound speed c > 0 ) . In other words, the classical solution obeys the generalized Huygens’ principle as that for the compressible Navier-Stokes equations in [5,6,15,27,28] due to the hyperbolic-parabolic structure of the system. In this direction, there are also similar space-time descriptions for the other compressible fluid models, for instance, the bipolar Navier-Stokes-Poisson equations [39,40]. However, we know that some compressible fluid models do not obey the generalized Huygens’ principle, such as the damped Euler equations [35,42] due to the damped mechanism and the unipolar Navier-Stokes-Poisson equations [36] due to the presence of the electronic field. Last but not least, we shall also refer to some work on the large time behavior of the other related models in [7,8,20,22,25,26,29,30,43–45] and references therein.

On the contrary, for the critical case P ′ ( ρ ¯ ) = 0 , Chikami and Kobayashi [3] recently established global existence of the strong solution in critical Besov space ( Λ ( ρ ˜ − ρ ¯ ) , m ) ∈ B ˙ 2,1 d 2 − 3 ( R d ) ∩ B ˙ 2,1 d 2 − 1 ( R d ) with d ≥ 3 and also showed the decay estimates of such global solutions under additional regularity assumption B ˙ 2 , ∞ − d 2 ( R d ) . This is the first global existence for this special case. To overcome the loss of lower order dissipation of the density arising from P ′ ( ρ ¯ ) = 0 , Chikami and Kobayashi [3] gave a subtle assumption in the low-frequency and fully used the conservative structure of the system, which is the reason why they considered the unknowns: the density ρ and the momentum m . Recently, Song and Xu [33] extended the result in [3] to the L p -critical Besov space.

We are interested in the classical solution for the Cauchy problem of system (1.1) for the special case P ′ ( ρ ¯ ) = 0 . In particular, we shall give the pointwise space-time description for this special case, which is inevitably different from the pointwise description given in [19]. To this end, we first prove that the classical solution globally exists. In fact, we need higher regularity to obtain the pointwise description of the solution when t ≥ 0 due to the quasi-linearity of the system and the singularity of Green’s function at t = 0 . Our global existence of the classical solution is partially inspired from [3], that is, we also need to impose some additional low-frequency assumption on the initial perturbation to overcome the loss of dissipation of the first derivative of the density ρ , and still consider the conservative variables ρ ˜ , m = ρ ˜ u with the initial data

More precisely, we are going to obtain the global solution for the small initial perturbation ( Λ ( ρ ˜ 0 − ρ ¯ ) , m 0 ) ∈ B ˙ 2,1 d 2 − 3 ( R d ) ∩ H k ( R d ) ( k ≥ [ d 2 ] + 1 ) with d ≥ 3 . Note that when d ≥ 6 B ˙ 2,1 d 2 − 3 ( R d ) ∩ H k ( R d ) = H k ( R d ) , that is, one can obtain the global existence of classical solution in a pure H k -framework when d ≥ 6 .

On the contrary, we shall obtain the optimal L 2 -decay rate of the smooth solution and its derivatives. In fact, Chikami and Kobayashi [3] have established the optimal decay rate of the strong solution in B ˙ 2,1 s ( R d ) with s ∈ ( − d 2 , d 2 + 1 ] and d ≥ 3 . Here, we derive the optimal L 2 -decay rate of the smooth solution and its derivatives by using two different methods. In fact, our L 2 -decay rate for the higher order derivatives of the smooth solution will be used to obtain the pointwise space-time behavior of the solution.

Finally, we provide the pointwise description of the solution by using Duhamel’s principle together with the energy estimates and L 2 -decay rate of the solution. We would like to state some differences in Green’s functions between the cases P ′ ( ρ ¯ ) > 0 and P ′ ( ρ ¯ ) = 0 (see the details in Section 2):

(1). The first difference is in G 22 for these two cases:

When P ′ ( ρ ¯ ) > 0 , there exist the operators with symbols cos ( c ∣ ξ ∣ t ) ξ ξ τ ∣ ξ ∣ 2 ( e − c 1 ∣ ξ ∣ 2 t − e − c 2 ∣ ξ ∣ 2 t ) and ( cos ( c ∣ ξ ∣ t ) − 1 ) e − ∣ ξ ∣ 2 t ξ ξ τ ∣ ξ ∣ 2 for positive constants c 1 and c 2 in G ˆ 22 , and they will produce so-called Riesz wave behaving as t − d 2 1 + ∣ x ∣ 2 t − d 2 (see (1.2)).

When P ′ ( ρ ¯ ) = 0 , although G ˆ 22 contains the terms like ( e − c 1 ∣ ξ ∣ 2 t − e − c 2 ∣ ξ ∣ 2 t ) ξ ξ τ ∣ ξ ∣ 2 , one can obtain the better estimate as the heat kernel t − d 2 e − ∣ x ∣ 2 t by using the cancellation.

(2). The second difference is in G 12 for these two cases:

When P ′ ( ρ ¯ ) > 0 , it holds that G ˆ 12 = − i e λ + t − e λ − t λ + − λ − ξ T ∼ i e − ∣ ξ ∣ 2 t sin ( c ∣ ξ ∣ t ) ∣ ξ ∣ ξ T , which yields that G 12 behaves as a Huygens’ wave in the low frequency as in (1.2) due to the fact λ ± ∼ i c ∣ ξ ∣ with the sound speed c = P ′ ( ρ ¯ ) .

When P ′ ( ρ ¯ ) = 0 , it holds that G ˆ 12 = − i e λ + t − e λ − t λ + − λ − ξ T ∼ i ξ T ∣ ξ ∣ 2 e − ∣ ξ ∣ 2 t , which yields that ∂ x α G 12 behaves as ( 1 + t ) − d − 1 + ∣ α ∣ 2 1 + ∣ x ∣ 2 1 + t − d − 1 + ∣ α ∣ 2 according to Lemma A.6.

Before stating our main result, we first introduce some notations which will be used frequently throughout the rest of the article.

Notation. C denotes a generic positive constant which may vary in different estimates. We write f ≲ g instead of f ≤ C g . We note that ∇ = ( ∂ x 1 , … , ∂ x d ) and for a multi-index α = ( α 1 , … , α d ) , ∂ x α = ∂ x 1 α 1 ∂ x 2 α 2 … ∂ x d α d , and ∣ α ∣ = ∑ i = 1 d α i . We also use ∂ x k to denote ∂ x α with ∣ α ∣ = k . The norm in the usual homogeneous and inhomogeneous Sobolev spaces H ˙ s ( R d ) and H s ( R d ) are denoted by ‖ ⋅ ‖ H ˙ s and ‖ ⋅ ‖ H s . In addition, let f ˆ ( ξ ) be the Fourier transform of f ( x ) with respect to x ∈ R d , that is, f ˆ ( ξ ) = ℱ ( f ) ( ξ ) = ( 2 π ) − d ⁄ 2 ∫ R d f ( x ) e − i x ⋅ ξ d x . We also define the fractional derivative operator by Λ s ≔ ( − Δ ) s ⁄ 2 = ℱ − 1 ∣ ⋅ ∣ s ℱ .

The main results in this article are stated as follows.

In the following, we always use f L and f H to denote the low- and high-frequency parts of a well-defined function f .

The rest of this article is organized as follows. In Section 2, we shall obtain the representation of Green’s function in Fourier space and give the pointwise estimate for Green’s function. The global existence and L 2 -decay rate of the solution are derived in Sections 3 and 4. In Section 5, we obtain the pointwise space-time behavior of the solution for the nonlinear problem. At last, some useful lemmas are stated in Appendix.

At the end of this section, we shall state the definition of homogeneous Besov space for completeness. We first recall some basic facts on the Littlewood-Paley theory (see [1] for instance). Let χ ≥ 0 and φ ≥ 0 be two smooth radial functions so that the support of χ is contained in the ball { ξ ∈ R d : ∣ ξ ∣ ≤ 4 3 } , the support of φ is contained in the annulus { ξ ∈ R d : 3 4 ≤ ∣ ξ ∣ ≤ 8 3 } and

The homogeneous dyadic blocks Δ ˙ j and the low-frequency cutoff operators S ˙ j are defined for all j ∈ Z by

Remark that for any homogeneous function η of order 0 smooth outside 0, we have

Denote by S h ′ ( R d ) the space of tempered distributions subject to the condition

Then, we have the decomposition

Now, we recall the definition of homogeneous Besov spaces.

2 Green’s function

In what follows, we assume that the steady state of the Cauchy problem (1.1)–(1.3) is ( ρ ¯ , 0 ) . For notational simplicity, we use ( ρ , m ) to denote the perturbation ( ρ ˜ − ρ ¯ , m ) . Then, system (1.1) can be rewritten in the perturbation form as

where the nonlinear term F 1 is defined as

Denote Green’s function of the linearized system for (2.1) by

where δ 0 is the Dirac- δ function and

A direct computation gives the representation of Green’s function in Fourier space:

when λ + ≠ λ − . Here,

As a result, one has

When λ + = λ − (or η ≔ 2 μ + μ ′ ρ ¯ = 2 κ ρ ¯ ), Green’s function has the following representation in the Fourier space:

Now, we shall state the differences in Green’s functions between the case P ′ ( ρ ¯ ) > 0 and the case P ′ ( ρ ¯ ) = 0 as follows:

First, when P ′ ( ρ ¯ ) > 0 , the lowest power of ∣ ξ ∣ in the imaginary part of λ ± for the low-frequency is one, and hence, there exists wave operators in the low frequency. As a result, one can deduce the Huygens’ wave in the low frequency and ultimately verify the generalized Huygens’ principle for the nonlinear problem. When P ′ ( ρ ¯ ) = 0 (the sound speed is zero), the lowest power of ∣ ξ ∣ in λ ± in the low frequency is two, and there is no wave operator in the low frequency. Hence, we can verify that the pointwise space-time description of Green’s function only contain the diffusion wave with zero sound speed (like the heat kernel).

Second, there exist the operators with symbols cos ( c ∣ ξ ∣ t ) ξ ξ τ ∣ ξ ∣ 2 ( e − a 1 ∣ ξ ∣ 2 t − e − a 2 ∣ ξ ∣ 2 t ) and ( cos ( c ∣ ξ ∣ t ) − 1 ) e − ∣ ξ ∣ 2 t ξ ξ τ ∣ ξ ∣ 2 for some positive constant a 1 and a 2 in G 22 when P ′ ( ρ ¯ ) > 0 , and this Riesz operator will produce the so-called Riesz wave behaving as t − d 2 1 + ∣ x ∣ 2 t − d 2 in the pointwise description of G 22 and hence affects the estimate for the solution of the nonlinear problem. However, when P ′ ( ρ ¯ ) = 0 , from G 22 in (2.5) and (2.8) and recalling (2.6)–(2.7), we find that although G 22 contains the terms like ( e − a 1 ∣ ξ ∣ 2 t − e − a 2 ∣ ξ ∣ 2 t ) ξ ξ τ ∣ ξ ∣ 2 , one can verify that it has better estimate as the heat kernel t − d 2 e − ∣ x ∣ 2 ⁄ C t by using the cancellation.

Third, G ˆ 12 will produce Huygens’ wave since the lowest power of ∣ ξ ∣ in λ ± in the low frequency is one when P ′ ( ρ ¯ ) > 0 , and there is no singularity based on the reformulation. However, for P ′ ( ρ ¯ ) = 0 , G ˆ 12 is of the form − i e λ + t − e λ − t λ + − λ − ξ T or − i t e − η 2 ∣ ξ ∣ 2 t ξ T . These two terms have the same time-decay rate. Nevertheless, the former is singular when ∣ ξ ∣ = 0 due to the operator with the symbol ξ ∣ ξ ∣ 2 . This operator is also contained in the electronic field ∇ ϕ = ∇ Δ ρ of the compressible Navier-Stokes-Poisson equations [36]. Thus, in virtue of Lemma A.6, the former term can be bounded by t − d − 1 2 1 + ∣ x ∣ 2 t − d − 1 2 .

In particular, we have the following pointwise description when P ′ ( ρ ¯ ) = 0 .

As a result, one can immediately obtain the following L 2 -decay estimate of the low-frequency of the solution for the linear Navier-Stokes-Korteweg system (2.1) when P ′ ( ρ ¯ ) = 0 .

This corollary will be used to obtain the optimal decay rate of any order derivative of the solution for the nonlinear problem.

3 Global existence

The global existence is usually derived by combining local existence with the a priori estimate as usual. In fact, the local existence can be established by a standard contraction mapping argument, and hence, we mainly focus on giving the a priori estimate. In this section, we shall consider the critical case: P ′ ( ρ ¯ ) = 0 . The main results are stated as follows.

The local existence was given in [3] when the initial perturbation is in B ˙ 2,1 d 2 − 1 ( R d ) for d ≥ 2 . In the same way, one can prove the local existence in the framework of Theorem 1.1. For simplicity, we omit the details. Thus, in the following, we only need to obtain the a priori estimate as follows: