On Regular Solutions for Three-Dimensional Full Compressible Navier–Stokes Equations with Degenerate Viscosities and Far Field Vacuum

Abstract

In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier–Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute temperature $\theta$ in a power law ($\theta ^\nu$ with $\nu >0$) of Chapman–Enskog, based on some elaborate analysis of this system’s intrinsic singular structures, we identify one class of initial data admitting a local-in-time regular solution with far field vacuum in terms of the mass density $\rho$, velocity u and entropy S. Furthermore, it is shown that within its life span of such a regular solution, the velocity stays in an inhomogeneous Sobolev space, i.e., $u\in H^3({\mathbb {R}}^3)$, S has uniformly finite lower and upper bounds in the whole space, and the laws of conservation of total mass, momentum and total energy are all satisfied. Note that, due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, and the physical entropy for polytropic gases behaves singularly, which make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the full CNS to the possible singularities of some special source terms related with S, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.

Appendix

For convenience of readers, we list some basic facts which have been used frequently in this paper.

The first one is the well-known Gagliardo–Nirenberg inequality.

Lemma 5.1

[23] Let function $u\in L^{q_1}\cap D^{1,r}({\mathbb {R}}^d)$ for $1 \le q_1, r \le \infty$. Suppose also that real numbers $\xi$ and $q_2$, and natural numbers m, i and j satisfy

$$\begin{aligned} \frac{1}{{q_2}} = \frac{j}{d} + \left( \frac{1}{r} - \frac{i}{d} \right) \xi + \frac{1 - \xi }{q_1} \quad \text {and} \quad \frac{j}{i} \le \xi \le 1. \end{aligned}$$

Then $u\in D^{j,{q_2}}({\mathbb {R}}^d)$, and there exists a constant C depending only on i, d, j, $q_1$, r and $\xi$ such that

$$\begin{aligned} \begin{aligned} \Vert \nabla ^{j} u \Vert _{L^{{q_2}}} \le C \Vert \nabla ^{i} u \Vert _{L^{r}}^{\xi } \Vert u \Vert _{L^{q_1}}^{1 - \xi }. \end{aligned} \end{aligned}$$

(5.1)

Moreover, if $j = 0$, $ir < d$ and $q_1 = \infty$, then it is necessary to make the additional assumption that either u tends to zero at infinity or that u lies in $L^s({\mathbb {R}}^d)$ for some finite $s > 0$; if $1< r < \infty$ and $i -j -d/r$ is a non-negative integer, then it is necessary to assume also that $\xi \ne 1$.

The second one concerns commutator estimates, which can be found in [33].

Lemma 5.2

[33] Let r, ${r_1}$ and ${r_2}$ be constants such that

$$\begin{aligned} \frac{1}{r}=\frac{1}{r_1}+\frac{1}{{r_2}},\quad \text {and} \quad 1\le r_1,\ {r_2}, \ r\le \infty . \end{aligned}$$

For $s\ge 1$, if $f, g \in W^{s,r_1} \cap W^{s,{r_2}}({\mathbb {R}}^3)$, then it holds that

$$\begin{aligned}{} & {} |\nabla ^s(fg)-f \nabla ^s g|_r\le C_s\big (|\nabla f|_{r_1} |\nabla ^{s-1}g|_{r_2}+|\nabla ^s f|_{r_2}|g|_{r_1}\big ), \end{aligned}$$

(5.2)

$$\begin{aligned}{} & {} |\nabla ^s(fg)-f \nabla ^s g|_r\le C_s\big (|\nabla f|_{r_1} |\nabla ^{s-1}g|_{r_2}+|\nabla ^s f|_{r_1}|g|_{r_2}\big ), \end{aligned}$$

(5.3)

where $C_s> 0$ is a constant depending only on s, and $\nabla ^s f$ ($s\ge 1$) is the set of all $\partial ^\varsigma _x f$ with $|\varsigma |=s$. Here $\varsigma =(\varsigma _1,\varsigma _2,\varsigma _3)^\top \in {\mathbb {R}}^3$ is a multi-index.

The last lemma is used to give some compactness results obtained via the Aubin-Lions Lemma.

Lemma 5.3

[41] Let $X_0\subset X\subset X_1$ be three Banach spaces. Suppose that $X_0$ is compactly embedded in X and X is continuously embedded in $X_1$. Then the following statements hold.

• i) If J is bounded in $L^r([0,T];X_0)$ for $1\le r < +\infty$, and $\frac{\partial J}{\partial t}$ is bounded in $L^1([0,T];X_1)$, then J is relatively compact in $L^r([0,T];X)$;

• ii) If J is bounded in $L^\infty ([0,T];X_0)$ and $\frac{\partial J}{\partial t}$ is bounded in $L^r([0,T];X_1)$ for $r>1$, then J is relatively compact in C([0, T]; X).