Blow-up criteria for the full compressible Navier-Stokes equations involving temperature in Vishik Spaces

1.   Introduction

We study the following system of Newton heat-conducting compressible fluid in three-dimensional space

Here, $\rho, \, u, \, \theta$ stand for the flow density, velocity and the absolute temperature, respectively. The scalar function $P$ represents the pressure, the state equation of which is determined by

and $\kappa$ is a positive constant and two constants $\mu$ and $\lambda$ are the coefficients of viscosity satisfying the physical restrictions $\mu > 0, \ 2\mu+3\lambda\ge0$. The initial conditions satisfy

Let $\gamma > 0$. For all $(t, x)\in \mathbb{R}\times \mathbb{R}^3$, we consider the following scaled functions:

There are huge literatures on the study of the existence of solutions to compressible Navier-Stokes equations, we only give a brief survey for blow-up criteria rather than the existence of solutions. When the initial data contain vacuums, after Xin's blow-up works ^[21,22], the various result for blow up critria for strong solutions to the system (1.1) is investigated. In present paper, in particular, we focus on the Serrin type criteria (e.g. ^[6,7,8,9]) as

or

and it is aimed to expand them into Vishik space motivated by the results of two recent papers Kanamaru ^[10] and Wu ^[20] (see also ^{[2,3,4,5,11,12,13,14,15,16,18,19]} for other criteria containing Beale-Kato-Majda blow-up mechanism).

We remind the local well-posedness of strong solutions to the equations (1.1) (see ^[1]).

Theorem 1.1. Let $\lambda < 3\mu$. Suppose $u_0, \theta_0 \in(D^1\cap D^2)(\mathbb{R}^{3})$ and $\rho_0 \in (W^{1, q}\cap H^1\cap L^1)(\mathbb{R}^{3})$ for some $q\in (3, 6]$. If $\rho_0$ isnonnegative and the initial data satisfy the compatibility condition

forvector fields $g_1, g_2\in L^2(\mathbb{R}^{3})$. Then there exist atime $T\in (0, \infty]$ and unique solution tp the equations (1.1)–(1.3), satisfying

If the maximal existence time $T^*$ is finite, then there holds

where $\sigma \in [1, \infty], \theta \in [1, \sigma]$.

Remark 1.1. In the light of the arguments in ^[7,8], we observe that (1.5) be replaced by

We note that the condition (1.5) is in scaling invariant norm in the sense of (1.4) for the temperature.

Remark 1.2. Without the restriction $\lambda < 3\mu$, in the case away from vacuum, through the argument in ^[9] and our proof, we obtain the similar results [9,Theorem 1.3] of what the authors in ^[9] says in Vishik space.

Next, we consider the full compressible Navier-Stokes equations without temperature.

where $\rho, u,$ and $P$ are the density, velocity and pressure respectively. The equation of state is given by

The constants $\mu$ and $\lambda$ are the shear viscosity and the bulk viscosity coefficients respectively. They satisfy the following physical restrictions: $\mu > 0, \quad 3\lambda + 2\mu\ge 0$.

Through a similar scheme in Theorem 1.1, we also obtain the following result for the equations (1.6).

Theorem 1.2. Let $(\rho, u)$ be a strongsolution to the Cauchy problem (1.6)–(1.7) with theinitial data $(\rho_0, u_0)$ satisfy

for some $r\in (3, \infty)$ and the compatibility condition:

If $T^* < \infty$ is the maximal time of existence, then both

and

where $\sigma \in [1, \infty], \theta \in [1, \sigma]$.

2.   Notations and some auxiliary lemmas

We follow the notation of ^[6] and ^[9]. For $1\leq p\leq\infty$, $L^{p}(\mathbb{R}^{3})$ represents the usual Lebesgue space. The classical Sobolev space $W^{k, p}(\mathbb{R}^{3})$ is equipped with the norm $\|f\|_{W^{k, p}(\mathbb{R}^{3})} = \sum\limits_{\alpha = 0}^{k}\|D^{\alpha}f\|_{L^{p}(\mathbb{R}^{3})}$. A function $f$ belongs to the homogeneous Sobolev spaces $D^{k, l}$ if $u\in L^1_{\text{loc}}(\mathbb{R}^3): \|\nabla^k u \|_{L^l} < \infty.$ $C > 0$ is an absolute constant which may be different from line to line unless otherwise stated in this paper. We also now introduce a Banach space $\dot{V}^s_{p, \sigma, \theta}(\mathbb{R}^3)$ which is larger than the homogeneous Besov space; see ^[10,17].

Definition 2.1. Let $s\in \mathbb R, p, \sigma \in [1, \infty], \theta \in [1, \sigma]$, the Vishik space $\dot{V}^s_{p, \sigma, \theta}$ is defined by

with the norm

Here $\mathcal{D}^\prime (\mathbb R^3)$ is the dual space of $\mathcal{D}(\mathbb R^3) = \{ f \in \mathcal{S}(\mathbb R^3); D^\alpha \hat{f} (0) = 0, \forall \ \alpha \in \mathbb{N}^3\}.$ As mentioned in ^[20], we remind that the following continuous embeddings hold:

for $s \in \mathbb R$, $p, \sigma \in [1, \infty]$ and $\theta_1, \theta_2 \in [1, \sigma]$ with $\theta_1 \geq \theta_2$.

In what follows, for simplicity, we write

3.   Proof of Theorem 1.1

We will prove Theorem 1.1 by a contradiction argument. Therefore, we assume that

Lemma 3.1. Suppose that $(3.1)$ is valid and $\lambda < 3\mu$, then there holds

Proof. From Lemma 2.3 and Lemma 3.1 in ^[9], we know that

and

Multiplying the inequality (3.3) by $(C +1)$ and adding the result to the inequality (3.2), we have

For the second term in the right hand side of (3.4), we note that

Now, let's control each term sequentially by Hölder's inequality, interpolation inequality(for the term $II$ below), Sobolev embedding theorem, Berstein's inequality and Young's inequalities:

(The term (I)):

(The term (II)):

(The term (III)):

Summing up the estimates above, we have

By similar above arguments, we get

Substituting (3.7) and (3.8) into (3.4), we obtain

where we used the fact that

for a sufficiently large constant $C > 0$. Now, choosing $N > 0$ sufficiently large such that $C2^{-N^2} \|\theta \|^2_{L^2} \leq \frac{1}{128}$, the estimate (3.9) becomes

Then, Grönwall's inequality and (3.10) enables us to obtain that the desired results.

Proof of Theorem 1.1. In the proof in Theorem 1.1 in ^[9], as long as Lemma 3.2 in ^[9] is only replaced by Lemma 3.1 in present paper, the proof is completed.

4.   Proof of Theorem 1.2

Let $(\rho, u)$ be a strong solution to the problem (1.6)-(1.7) as described in Theorem 1.2. Then the standard energy estimate yields

We first prove Theorem 1.2 by a contradiction argument. Otherwise, there exists some constant $M_0 > 0$ such that

The first key estimate on $\nabla u$ will be given in the following lemma.

Lemma 4.1. Under the condition (4.2), it holds that

Proof. It follows from the momentum equations in (1.6) that

where $\dot{v}: = v_t+u\cdot\nabla v, \quad G: = (2\mu + \lambda)\text{div}u - P(\rho), \quad{\omega}: = \nabla \times u$ are the material derivative of $f,$ the effective viscous flux $G$ and the vorticity ${\omega}$, respectively. In particular, for the effective viscous flux, it is well-known that

and

Multiplying the momentum equation $(1.6)_2$ by $u_t$ and integrating the resulting equation over $\mathbb{R}^3$ gives

From $(1.6)_1$, we note that

For the first term in the right hand side of (4.5), one has

Substituting (4.6) into (4.5), we have

For the second term in the right hand side of (4.5), we have

In a similar way to (3.5), we let control each term sequentially.

(The term (I)):

(The term (II)):

(The term (III)):

Summing up the estimates $I$–$III$, it is bounded by

On the other hand, due to (4.4), we note that

Collecting (4.7) and (4.8), we have

Now, choosing $N > 0$ sufficiently large such that $C2^{-N^2} \|u \|^2_{L^2} \leq \frac{1}{128}$, (indeed, the constant $C > 0$ is also depending on $\|\rho_0^{1/2} u_0\|^2_{L^2}$) the estimate (4.9) becomes

which, together with (4.2) and Grönwall's inequality, gives (4.3). The proof of Lemma 4.1 is completed.

Proof of Theorem 1.2. In the proof in Theorem 1.1 in ^[6], as long as Lemma 3.1 in ^[6] is only replaced by Lemma 4.1 in our paper, the proof is completed.

5.   Appendix

For the convenience of the reader, we give the proof for (4.6), given in ^[6].

6.   Discussion

Our result is focused on the full compressible Navier-Stokes equationss. However, it is believed that our results can be expanded in various ways for the coupled equations or system. In this regard, we think of it as a future study and intend to produce more meaningful results.

7.   Conclusions

The current paper results are Blow-up criteria for solutions in Vishik Space which is a weaker space to Besov space and Lebesgue space. It seems to be a meaningful result in this regard, and it is new.

Acknowledgments

The authors thank the very knowledgeable referee very much for his/her valuable comments and helpful suggestions. Jae-Myoung Kim was supported by National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521).

Conflict of interest

The authors declare that there are no conflicts of interest in this paper.