Serrin-type blowup Criterion for the degenerate compressible Navier-Stokes equations

1.   Introduction

The motion of a general viscous isentropic compressible fluid occupying a spatial domain $\Omega\subset \mathbb{R}^3$ can be described by the following isentropic compressible Navier–Stokes equations:

where $\rho\geq 0$ is the mass density; $u = (u_1, u_2, u_3)$ is the velocity of the fluid; $x = (x_1, x_2, x_3)\in \Omega$, $t\geq 0$ are the space and time variables, respectively. For the polytropic gases, the constitutive relation, which is also called the equations of state, is given by

Here $P$ is the pressure, $A > 0$ is an entropy constant and $\gamma$ is the adiabatic exponent. $\mathbb{T}$ denotes the viscous stress tensor with the form

where $\mathbb{I}_3$ is the $3\times 3$ identity matrix,

for some constant $\delta\geq 0$, $\mu(\rho)$ is the shear viscosity coefficient, $\lambda(\rho)+\frac{2}{3}\mu(\rho)$ is the bulk viscosity coefficient, $\alpha$ and $\beta$ are both constants satisfying

This system can be derived from the Boltzmann equations through the Chapman–Enskog expansion, cf. Chapman–Cowling ^[1] and Li–Qin ^[2].

In this paper, we consider the Cauchy problem for (1.1) with the following initial data and far-field behavior:

In addition, we will use the following simplified notations; most of them are introduced in the standard homogeneous and inhomogeneous Sobolev spaces ^[3]:

For the initial data away from vacuum, the local existence of classical solutions for (1.1)-(1.6)-(1.7) is proved due to the standard symmetric hyperbolic-parabolic structure which satisfies the well-known Kawashima's condition ^[4,5]. However, for the initial data including vacuum, such an approach does not work because of the degeneration in the momentum equation. Generally, a vacuum often occurs in some physical requirements, such as the finite total initial mass and the finite total initial energy. It is well known that the main difficulty is to understand the behavior of the velocity field near the vacuum.

When the coefficients of viscosity $\mu$ and $\lambda$ are constant $(\delta = 0)$, the degeneration occurs only in the time evolution of the momentum equation, Cho–Choe–Kim ^[6] introduce a remedy by this initial compatibility condition:

which implies that $(\sqrt{\rho}u_t, \nabla u_t)$ in $L^{\infty}([0, T^{*}]; L^2)$ for a short time $T^{*}$. Then they obtain successfully the local well-posedness of smooth solutions with vacuum in some three-dimensional Sobolev spaces, and also show the necessity of the initial compatibility condition in their solution class. And Huang, Li, and Xin ^[7] establish the global well-posedness of these classical solutions with small energy and vacuum. For some related results, please refer to ^{[8,9,10,11,12,13,14,15]}.

When the coefficients of viscosity $\mu$ and $\lambda$ are dependent on the density $(\delta > 0)$, the system (1.1) has received a lot of attention, see ^{[16,17,18,19,20,21,22]}. However, both the time evolution and viscosities are degenerate near the vacuum in the momentum equation, which prevents us from utilizing a similar remedy proposed in ^[6]. Recently, Zhu and his collaborators obtained some important advances on the well-posedness of classical solutions in this case; see ^{[23,24,25,26,27,28,29]}. Based on this observation for the momentum equation in (1.1), this one can also be rewritten in the following form:

where

As $\delta = 1$, they ^[24] introduce uninformative a priori estimates of $\nabla \log \rho$ in $L^6\cap D^1\cap D^2$ to establish the existence of a 2-D local classical solution with far-field vacuum, which is also extended to the three-dimensional spaces in ^[29]. As $\delta\in (1, +\infty)$, by using some hyperbolic approaches when $\rho > 0$ in (1.8) and the hyperbolic one

when $\rho = 0$, they ^[25] establish the existence of a 3-D local classical solution with a vacuum. The corresponding global well-posedness under some initial smallness assumptions is also established in ^[26]. As $\delta\in (0, 1)$, they ^[27] introduce an elaborate elliptic approach on the operators $L(\rho^{\delta-1}u)$ and some initial compatibility conditions, to obtain the well-posedness of a local regular solution with far-field vacuum in some inhomogeneous Sobolev spaces.

In the current paper, we are concerned with the main mechanism for possible breakdown of classical solutions for the Cauchy problem (1.1)-(1.6)-(1.7) with $\delta > 1$ obtained in ^[25]. Our result shows that, the $L^\infty$ norms of $\nabla u, \nabla \rho^{\frac{\delta-1}{2}}\, \text{and} \, \nabla \rho^{\frac{\gamma-1}{2}}$ control the possible breakdown of this solution, which means that if a solution of (1.1)-(1.6)-(1.7) is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $\nabla u$, $\nabla \rho^{\frac{\delta-1}{2}}$ or $\nabla \rho^{\frac{\gamma-1}{2}}$ as the critical time approaches; equivalently, if they all remain bounded, a regular solution persists. This conclusion can be stated precisely as follows.

Theorem 1.1. If $(\rho, u)$ is the unique classical solution obtained in Theorem 2.1, and $\overline{T} < +\infty$ is the maximal existence time of $(\rho, u)$, then

Furthermore, if $\gamma\le \delta$, then

This paper is organized as follows. In §2, we introduce some known well-posedenss theories of the Cauchy problem (1.1)-(1.6)-(1.7) and some fundamental lemmas; in §3, we provide the detailed proof of Theorem 1.1.

2.   Preliminaries

In this section, we introduce some known well-posedness theories of the Cauchy problem (1.1)-(1.6)-(1.7) and some fundamental lemmas that are frequently used in our proof.

In order to state our results clearly, we introduce the following regular solutions of the Cauchy problem (1.1)-(1.6)-(1.7) from ^[23].

Definition 2.1. (Regular solutions). ^[23] Let $T > 0$ be a positive time. The function pair $(\rho(t, x), u(t, x))$ is called a regular solution to the Cauchy problem (1.1)-(1.6)-(1.7) in $[0, T] \times \mathbb{R}^3$ if $(\rho(t, x), u(t, x))$ satisfies this problem in the sense of distributions and:

where $s'\in [2, 3)$ is an arbitrary constant.

The well-posedness of these regular solutions has been established in ^[23] as follow:

Theorem 2.1. ^[23] Assume $\delta > 1$ in (1.4). If initial data $(\rho_0, u_0)$ satisfy

then there exists a time $T_* > 0$, and a unique regular solution $(\rho, u)$ in $[0, T_*]\times \mathbb{R}^3$ to the Cauchy problem (1.1)-(1.6)-(1.7) satisfying

where $s'\in [2, 3)$ is an arbitrary constant. Moreover, if $1 < \min\{\gamma, \delta\}\leq 3$, $(\rho, u)$ is indeed a classical solution of the Cauchy problem (1.1)-(1.6)-(1.7) in $(0, T_*]\times \mathbb{R}^3$.

Next, we introduce the well-known Gagliardo–Nirenberg inequality and Moser-type calculus inequality.

Lemma 2.1. ^[30] Let function $u\in L^q\cap D^{1, r}(\mathbb{R}^n)$ for $1 \leq q, r \leq \infty$. Suppose also that a real number $\theta$ and natural numbers $m$ and $j$ satisfy

Then $u\in D^{j, p}(\mathbb{R}^n)$, and there exists a constant $C$ depending only on $m$, $n$, $j$, $q$, $r$ and $\theta$ such that

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

Lemma 2.2. ^[31] Let $r$, $a$ and $b$ be constants such that

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

where $C_s > 0$ is a constant depending only on $s$, and $\nabla^s f$ ($s\geq 1$) is the set of all $\partial^\zeta_x f$ with $|\zeta| = s$. Here $\zeta = (\zeta_1, ..., \zeta_n)\in \mathbb{R}^n$ is a multi-index.

3.   Serrin-type blow-up criterion

The purpose of this section is to prove Theorem 1.1. Let $(\rho, u)$ be the classical solution to (1.1)-(1.6)-(1.7) obtained in Theorem 2.1 in $[0, T]\times \mathbb{R}^3$. Suppose that $\overline{T}$ is the maximal existence time, and the opposite of (1.11) holds, i.e.,

where $\bar{c} > 0$ is some finite constant.

3.1.   Reformulation

By the two new quantities:

we can rewrite system (1.1) into a new system, which consists of a "quasi-symmetric hyperbolic"-"degenerate elliptic" coupled system with some special lower-order source terms for $(\phi, u)$, and a transport equation for $\varphi$:

where $W = (\phi, u)^\top$ and

with $a_1 = \frac{(\gamma-1)^2}{4A\gamma} > 0$ and $Q(u) = \frac{\delta}{\delta-1}\mathbb{S}(u)$. Meanwhile, $\partial_j W = \partial{x_j}W$, and

Here $e_j = (\delta_{1j}, \delta_{2j}, \delta_{3j})$ $(j = 1, 2, 3)$ is the Kronecker symbol satisfying $\delta_{ij} = 1$, when $i = j$ and $\delta_{ij} = 0$, otherwise. For any $\xi\in {\mathbb R}^4$, we have

Moreover,

3.2.   Uniform estimates for arbitrarily large time

Based on (3.1), we will make some uniform estimates for arbitrarily large times.

Lemma 3.1. If (3.1) holds, then

for any $0 < T\leq \overline{T}$, where the constant $C > 0$ is only dependent on $(\rho_0, u_0)$, $\bar{c}$, $\alpha$, $\beta$, $A$, $\gamma$ and $\delta$.

Proof. First, it is obvious that $\rho$ can be represented by

where $\Phi\in C^1([0, T]\times[0, T]\times \mathbb{R}^3)$ is the solution to the initial value problem

Then it is easy to derive that

□

Next, we are going to establish the $H^3$ estimates of $(\phi, u, \varphi)$. Let

for three multi-indexes $\zeta^i\in \mathbb{R}^3$ $(i = 1, 2, 3)$ satisfying $|\zeta^i| = 0$ or $1$. On the one hand, we apply the operator $\partial^\zeta_x$ to $(3.2)_1$, multiply the resulting equations by $\partial^\zeta_x W$ on both sides and integrate over $\mathbb{R}^3$ to have

On the other hand, we apply the operator $\partial^\zeta_x$ to $(3.2)_2$, multiply the resulting equations by $\partial^\zeta_x \varphi$ on both sides and integrate over $\mathbb{R}^3$ to have

Some terms in (3.11) and (3.12) are easy to be estimated in the following lemma.

Lemma 3.2. For $|\zeta| = k$, $k = 0, 1, 2, 3$, one has

where the constant $C > 0$ is only dependent on $(\rho_0, u_0)$, $\bar{c}$, $\alpha$, $\beta$, $A$, $\gamma$ and $\delta$.

Proof. For $|\zeta| = k = 0, 1, 2, 3$, from the Hölder and Sobolev inequalities and Lemma 2.2 we can derive that

which implies the desired conclusion.

□

Based on the relations (3.11)-(3.12) and Lemma 3.2, we can obtain the following lower order estimates.

Lemma 3.3. If $(\rho, u)$ satisfies (3.1), then it holds that for any $0 < T < \overline{T}$

where the constant $C > 0$ is only dependent on $(\rho_0, u_0)$, $\bar{c}$, $\alpha$, $\beta$, $A$, $\gamma$ and $\delta$.

Proof. Step 1. First, for the $L^2$-estimate, from (3.11)-(3.12), we can see that

By (3.16), (3.1), Young's inequality, and Grönwall's inequality, we can conclude that

Step 2. Second, for the $D^1$-estimate, i.e., $|\zeta| = 1$, we have

Based on (3.17), (3.1), Lemma 3.2, Young's inequality, and Grönwall's inequality, we can conclude that

□

The higher-order estimates for $W$ are also obtained as follow:

Lemma 3.4. If $(\rho, u)$ satisfies (3.1), then it holds that for any $0 < T < \overline{T}$

where the constant $C > 0$ is only dependent on $(\rho_0, u_0)$, $\bar{c}$, $\alpha$, $\beta$, $A$, $\gamma$ and $\delta$.

Proof. Step 1. First, for the $D^2$-estimate, i.e., $|\zeta| = 2$, we have

where the terms $J^*_3$ and $J^*_5$, via integration by parts, can be estimated as

for some multi-indexe $\zeta_*\in \mathbb{R}^3$ satisfying $|\zeta_*| = 1$.

Thus, by (3.11)-(3.12), Lemma 3.2, (3.19)-(3.20), and Young's inequality, we can conclude that

Step 2. Second, for the $D^3$-estimate, i.e., $|\zeta| = 3$, it follows from the Hölder and Sobolev inequalities that

where we have used the Gagliardo–Nirenberg inequality in Lemma 2.1:

and the term $J^{**}_3$, via integration by parts, can be estimated as

for some multi-indexes $\zeta_{**}\in \mathbb{R}^3$ satisfying $|\zeta_{**}| = 2$.

For the term $J_{5}$, we have

where the term $J^{**}_{5i}$ ($i = 1, 2, 3$), via integration by parts, can be estimated as

for some multi-indexes $\zeta_{*}, \zeta_{**}\in \mathbb{R}^3$ satisfying $|\zeta_{*}| = 1$ and $|\zeta_{**}| = 2$. For the last term in $J^{**}_{53}$, we still need the integration by parts to estimate it,

For the term $J_8$, we have

Then, according to (3.11)-(3.12), Lemma 3.1-3.2, (3.22)-(3.27), and Young's inequality, we can conclude that

which, together with Grönwall's inequality, (3.1) and (3.20), gives (3.18). Hence the proof is finished. □

3.3.   Proof of Theorem 1.1

Now, we know that if the classical solution $(\rho, u)(t, x)$ exists up to the time $\overline{T}$, where $\overline{T} < +\infty$ is the maximal existence time such that the assumption (3.1) holds, then we have Lemmas 3.1-3.4. Thus, by the standard weak compactness theory, we can see that for any sequence $\{t_k\}_{k = 1}^{\infty}$ with $t_k\in(0, \overline{T})$ and $t_k\rightarrow \overline{T}(k\rightarrow \infty)$, there exists one subsequence $\{t_{n_k}\}_{k = 1}^{\infty}$ and functions $(\phi, u, \varphi)(\overline{T}, x)$ such that

which implies that $(\phi, u, \varphi)(\overline{T}, x)$ satisfies the initial data (2.1) in Theorem 2.1. Thus, by Theorem 2.1, the classical solution $(\rho, u)$, can be extended beyond $[0, \overline{T}]$. This contradicts to the fact that $\overline{T}$ is the maximal existence time. Thus, we obtain (1.11).

Furthermore, when $\gamma\le \delta$, by Lemma 3.1 we have

Due to

it is obvious that

which implies that we can obtain (1.12) similar to (1.11). Therefore, we complete the proof of Theorem 1.1.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research is supported by the Outstanding Youth Research Project of Anhui Universities (2023AH030075); the Research and Innovation Team Project of Fuyang Normal University (kytd202003).

Conflict of interest

The authors declare there is no conflict of interest.