Existence of Dissipative (and Weak) Solutions for Models of General Compressible Viscous Fluids with Linear Pressure

Abstract

In this work we will focus on the existence of dissipative solutions for a system describing a general compressible viscous fluid in the case of the pressure being a linear function of the density and the viscous stress tensor being a non-linear function of the symmetric velocity gradient. Moreover, we will study under which conditions it would be possible to get the existence of weak solutions.

De la Vallée–Poussin Criterion

In this section, we prove a slightly modified version of the De la Vallée–Poussin criterion as we require the stronger condition, with respect to the standard formulation, that the Young function satisfies the $\Delta _2$-condition. We first recall the definitions of Young function and $\Delta _2$-condition.

Definition A.1

• (i) We say that $\Phi$ is a Young function generated by $\varphi$ if

  $$\begin{aligned} \Phi (t) = \int _{0}^{t} \varphi (s) \ {\mathrm{d}}s \quad \text{ for } \text{ any } t\ge 0, \end{aligned}$$

  where the real-valued function $\varphi$ defined on $[0,\infty )$ is non-negative, non-decreasing, left-continuous and such that

  $$\begin{aligned} \varphi (0)=0, \quad \lim _{s\rightarrow \infty }\varphi (s)=\infty . \end{aligned}$$

• (ii) A Young function $\Phi$ is said to satisfy the $\Delta _2$-condition if there exist a positive constant K and $t_0 \ge 0$ such that

  $$\begin{aligned} \Phi (2t) \le K \Phi (t) \quad \text{ for } \text{ any } t\ge t_0. \end{aligned}$$

  (A.1)

Theorem A.2

Let $Q \subset {\mathbb {R}}^d$ be a bounded measurable set and let $\{ f_n \}_{n\in {\mathbb {N}}}$ be a sequence in $L^1(Q)$. Then, the following statements are equivalent.

• (i) The sequence $\{ f_n \}_{n\in {\mathbb {N}}}$ is equi-integrable, meaning that for any $\varepsilon >0$ there exists $\delta =\delta (\varepsilon )>0$ such that

  $$\begin{aligned} \int _{M} |f_n (y)| \ {\mathrm{d}}y< \varepsilon \quad \text{ for } \text{ any } M\subset Q \text{ such } \text{ that } |M| < \delta , \end{aligned}$$

  independently of n.

• (ii) There exists a Young function $\Phi$ satisfying the $\Delta _2$-condition (A.1) such that the sequence $\{ f_n \}_{n\in {\mathbb {N}}}$ is uniformly bounded in the Orlicz space $L_{\Phi }(Q)$.

Proof

(ii) $\Rightarrow$ (i) See Pedregal [18], Chapter 6, Lemma 6.4.

(i) $\Rightarrow$ (ii) For $n\in {\mathbb {N}}$ and $j\ge 1$ fixed, let

$$\begin{aligned} \mu _j(f_n):= |\{ y\in Q: \ |f_n(y)| >j \}|. \end{aligned}$$

As the sequence $\{ f_n \}_{n\in {\mathbb {N}}}$ is equi-integrable, from the Dunford-Pettis theorem there exists a strictly increasing sequence of positive integers $\{ C_m \}_{m\in {\mathbb {N}}}$ such that for each m

$$\begin{aligned} \sup _{n\in {\mathbb {N}}} \int _{\{ |f_n|>C_m\}} |f_n(y)| \ {\mathrm{d}}y \le \frac{1}{2^m}. \end{aligned}$$

For $n\in {\mathbb {N}}$ and $m\ge 1$ fixed

$$\begin{aligned} \int _{\{ |f_n|>C_m\}} |f_n(y)| \ {\mathrm{d}}y = \sum _{j=C_m}^{\infty } \int _{\{ j<|f_n|\le j+1\}} |f_n(y)| \ {\mathrm{d}}y \ge \sum _{j=C_m}^{\infty } j \ [\mu _j(f_n)-\mu _{j+1}(f_n)] \ge \sum _{j=C_m}^{\infty }\mu _j(f_n). \end{aligned}$$

In particular, we obtain

$$\begin{aligned} \sum _{m=1}^{\infty } \sum _{j=C_m}^{\infty }\mu _j(f_n) \le \sum _{m=1}^{\infty } \int _{\{ |f_n|>C_m\}} |f_n(y)| \ {\mathrm{d}}y \le \sum _{m=1}^{\infty } \frac{1}{2^m} =1. \end{aligned}$$

For $m\ge 0$, we define

$$\begin{aligned} \alpha _m= {\left\{ \begin{array}{ll} 0 &{}\quad \text{ if } m<C_1, \\ \max \{k: \ C_k \le m\} &{}\quad \text{ if } m\ge C_1. \end{array}\right. } \end{aligned}$$

Notice that

$$\begin{aligned} \alpha _m \ge j \quad \Leftrightarrow \quad C_j \le m. \end{aligned}$$

(A.2)

It is straightforward that $\alpha _m \rightarrow \infty$ as $m\rightarrow \infty$. We define a step function $\varphi$ on $[0,\infty )$ by

$$\begin{aligned} \varphi (s) = \sum _{m=0}^{\infty } \alpha _m \chi _{(m,m+1]}(s) \quad \text{ for } \text{ any } 0\le s<\infty . \end{aligned}$$

It is clear that $\varphi$ is non-negative, non-decreasing, left-continuous and such that $\varphi (0)=0$, $\lim _{s\rightarrow \infty } \varphi (s)= \infty$. Then, we can define the Young function $\Phi$ generated by $\varphi$ as

$$\begin{aligned} \Phi (t)= \int _{0}^{t} \varphi (s)\ {\mathrm{d}}s, \quad \text{ for } \text{ any } 0\le t< \infty . \end{aligned}$$

At this point, notice that we have the freedom to take the constants $C_j$, $j\ge 1$, as large as we want and consequently, the constants $\alpha _m$, $m\ge 1$, will be as small as we want. More precisely, we may find a positive constant c such that

$$\begin{aligned} \alpha _{2m} \le c \ \alpha _m \quad \text{ for } \text{ any } m\ge 1. \end{aligned}$$

We then obtain, for all $s\in [0,\infty )$,

$$\begin{aligned} \varphi (2s) = \sum _{m=0}^{\infty } \alpha _m \chi _{\left( \frac{m}{2}, \frac{m+1}{2}\right) }(s) = \sum _{k=0}^{\infty } \alpha _{2k} \chi _{\left( k, k+ \frac{1}{2}\right) }(s) \le c \sum _{k=0}^{\infty } \alpha _k \chi _{\left( k, k+ \frac{1}{2}\right) }(s) \le c\ \varphi (s); \end{aligned}$$

consequently, for all $t\in [0,\infty )$,

$$\begin{aligned} \Phi (2t) = \int _{0}^{2t} \varphi (s) \ {\mathrm{d}}s = 2\int _{0}^{t} \varphi (2z) \ {\mathrm{d}}z \le 2c \int _{0}^{t} \varphi (z) \ {\mathrm{d}}z = 2c \ \Phi (t), \end{aligned}$$

and thus we get that the Young function $\Phi$ satisfies the $\Delta _2$-condition (A.1).

Finally, for $n\in {\mathbb {N}}$ fixed, using the fact that $\Phi (0)=\Phi (1)=0$ and for $j\ge 1$, noticing that $\alpha _0=0$,

$$\begin{aligned} \Phi (j+1) = \int _{0}^{j+1} \varphi (s) \ {\mathrm{d}} s = \sum _{m=0}^{j} \int _{m}^{m+1} \varphi (s) \ {\mathrm{d}} s \le \sum _{m=0}^{j} \varphi (m+1) = \sum _{m=0}^{j} \alpha _m= \sum _{m=1}^{j} \alpha _m, \end{aligned}$$

we get

$$\begin{aligned} \int _{Q} \Phi (|f_n(y)|) \ {\mathrm{d}} y&= \int _{\{ |f_n|=0 \}} \Phi (|f_n(y)|) \ {\mathrm{d}} y + \sum _{j=0}^{\infty } \int _{\{ j<|f_n|\le j+1\}} \Phi (|f_n(y)|) \ {\mathrm{d}}y \\&\le \sum _{j=1}^{\infty } [\mu _j(f_n)-\mu _{j+1}(f_n)] \ \Phi (j+1) \\&\le \sum _{j=1}^{\infty } [\mu _j(f_n)-\mu _{j+1}(f_n)] \sum _{m=1}^{j} \alpha _m \\&= \sum _{m=1}^{\infty } \alpha _m \sum _{j=m}^{\infty } [\mu _j(f_n)-\mu _{j+1}(f_n)] \\&= \sum _{m=1}^{\infty } \alpha _m \mu _m(f_n)= \sum _{m=1}^{\infty } \mu _m(f_n) \sum _{j=1}^{\alpha _m} 1 = \sum _{j=1}^{\infty } \sum _{m=C_j}^{\infty } \mu _m(f_n) \le 1 \end{aligned}$$

where we used (A.2) in the last line. In particular, we obtain that the sequence $\{ f_n \}_{n\in {\mathbb {N}}}$ is uniformly bounded in the Orlicz space $L_{\Phi }(Q)$. $\square$