On the Breakdown of Regular Solutions with Finite Energy for 3D Degenerate Compressible Navier–Stokes Equations

Abstract

In this paper, the three-dimensional (3D) isentropic compressible Navier–Stokes equations with degenerate viscosities (ICND) is considered in both the whole space and the periodic domain. First, for the corresponding Cauchy problem, when shear and bulk viscosity coefficients are both given as a constant multiple of the density’s power ($\rho ^\delta$ with $0<\delta <1$), based on some elaborate analysis of this system’s intrinsic singular structures, we show that the $L^\infty$ norm of the deformation tensor D(u) and the $L^6$ norm of $\nabla \rho ^{\delta -1}$ control the possible breakdown of regular solutions with far field vacuum. This conclusion means that if a solution with far field vacuum of the ICND system is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of D(u) or $\nabla \rho ^{\delta -1}$ as the critical time approaches. Second, when $0<\delta \le 1$, under the additional assumption that the shear and second viscosities (respectively $\mu (\rho )$ and $\lambda (\rho )$) satisfy the BD relation $\lambda (\rho )=2(\mu '(\rho )\rho -\mu (\rho ))$, if we consider the corresponding problem in some periodic domain and the initial density is away from the vacuum, it can be proved that the possible breakdown of classical solutions can be controlled only by the $L^\infty$ norm of D(u). It is worth pointing out that, except the conclusions mentioned above, another purpose of the current paper is to show how to understand the intrinsic singular structures of the fluid system considered now, and then how to develop the corresponding nonlinear energy estimates in the specially designed energy space with singular weights for the unique regular solution with finite energy.

Appendix: Some Basic Lemmas

In this section, we list some basic lemmas to be used later. The first one is the well-known Gagliardo–Nirenberg inequality.

Lemma 5.8

[26] For $p\in [2,6]$, $q\in (1,\infty )$, and $r\in (3,\infty )$, there exists some generic constant $C> 0$ that may depend on q and r such that for

$$\begin{aligned} f\in H^1(\mathbb {R}^3),\quad \text {and} \quad g\in L^q(\mathbb {R}^3)\cap D^{1,r}(\mathbb {R}^3), \end{aligned}$$

it holds that

$$\begin{aligned} \begin{aligned}&|f|^p_p \le C |f|^{(6-p)/2}_2 |\nabla f|^{(3p-6)/2}_2,\quad |g|_\infty \le C |g|^{q(r-3)/(3r+q(r-3))}_q |\nabla g|^{3r/(3r+q(r-3))}_r. \end{aligned} \end{aligned}$$

(5.33)

Some special cases of this inequality are

$$\begin{aligned} \begin{aligned} |u|_6\le C|u|_{D^1},\quad |u|_{\infty }\le C|u|^{\frac{1}{2}}_6|\nabla u|^{\frac{1}{2}}_{6}, \quad |u|_{\infty }\le C\Vert u\Vert _{W^{1,r}}. \end{aligned} \end{aligned}$$

(5.34)

The second lemma gives some compactness results obtained via the Aubin-Lions Lemma.

Lemma 5.9

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

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

The third one can be found in Majda [34].

Lemma 5.10

[34] Let r, a and b be constants such that

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

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

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

(5.35)

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

(5.36)

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

The following lemma is important in the derivation of the a priori estimates in Sect. 3, which can be found in Remark 1 of [2].

Lemma 5.11

[2] If $f(t,x)\in L^2([0,T]; L^2)$, then there exists a sequence $s_k$ such that

$$\begin{aligned} s_k\rightarrow 0, \quad \text {and}\quad s_k |f(s_k,x)|^2_2\rightarrow 0, \quad \text {as} \quad k\rightarrow +\infty . \end{aligned}$$

Next we give one Sobolev inequalities on the interpolation estimate in the following lemma.

Lemma 5.12

[34] Let $u\in H^s$, then for any $s'\in [0,s]$, there exists a constant $C_s$ only depending on s such that

$$\begin{aligned} \Vert u\Vert _{s'} \le C_s \Vert u\Vert ^{1-\frac{s'}{s}}_0 \Vert u\Vert ^{\frac{s'}{s}}_s. \end{aligned}$$

In order to improve a weak convergence to the strong convergence, we give the following lemma.

Lemma 5.13

[34] If the function sequence $\{w_n\}^\infty _{n=1}$ converges weakly to w in a Hilbert space X, then it converges strongly to w in X if and only if

$$\begin{aligned} \Vert w\Vert _X \ge \lim \text {sup}_{n \rightarrow \infty } \Vert w_n\Vert _X. \end{aligned}$$

The next lemma is used to give the estimate on $\nabla u_t$ in the periodic problem away from the vacuum.

Lemma 5.14

[11] Let $\Omega \subset \mathbb {R}^n$ $(n\ge 2)$ be an open, connected domain. Then there is a constant $C>0$, known as the Korn constant of $\Omega$, such that, for all vector fields $v=(v^1,\ldots , v^n)\in H^1(\Omega )$,

$$\begin{aligned} \Vert v\Vert ^2_{H^1(\Omega )}\le C\int _{\Omega } \big (|v|^2+|D(v)|^2\big ) \mathrm{d}x. \end{aligned}$$

Finally, we give the well-known Fatou’s lemma.

Lemma 5.15

Given a measure space $(V,\mathcal {F},\nu )$ and a set $X\in \mathcal {F}$, let $\{f_n\}$ be a sequence of $(\mathcal {F}, \mathcal {B}_{\mathbb {R}_{\ge 0}} )$-measurable non-negative functions $f_n: X\rightarrow [0,\infty ]$. Define the function $f: X\rightarrow [0,\infty ]$ by setting

$$\begin{aligned} f(x)= \liminf _{n\rightarrow \infty } f_n(x), \end{aligned}$$

for every $x\in X$. Then f is $(\mathcal {F}, \mathcal {B}_{\mathbb {R}_{\ge 0}})$-measurable, and

$$\begin{aligned} \int _X f(x) \mathrm{d}\nu \le \liminf _{n\rightarrow \infty } \int _X f_n(x) \mathrm{d}\nu . \end{aligned}$$