Local-in-time existence of strong solutions to a class of the compressible non-Newtonian Navier–Stokes equations

Abstract

The aim of this article is to show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equations for arbitrarily large initial data. The goal is reached by $L^p$-theory for linearized equations which are obtained with help of the Weis multiplier theorem and can be seen as a generalization of the work of Enomoto and Shibata (Funkcial Ekvac 56(3):441–505, 2013) (devoted to compressible fluids) to compressible non-Newtonian fluids.

Appendix

Here we present several theorems from other sources for readers convenience.

Theorem 4.1

(Theorem 3.3 in [13]) Let $1<q<\infty$ and let $\Lambda \subset {\mathbb {C}}$. Let $m(\lambda ,\xi )$ be a function defined on $\Lambda \times (\mathbb R^d{\setminus }\{0\})$ such that for any multi-index $\alpha \in {\mathbb {N}}^d_0$ there exists a constant $C_\alpha$ depending on $\alpha$ and $\Lambda$ such that

$$\begin{aligned} |\partial ^\alpha _\xi m(\lambda ,\xi )|\le C_\alpha |\xi |^{-\alpha } \end{aligned}$$

for any $(\lambda ,\xi )\in \Lambda \times ({\mathbb {R}}^d{\setminus } \{0\})$. Let $K_\lambda$ be an operator defined by $K_\lambda f = {\mathcal {F}}^{-1}_\xi [m(\lambda ,\xi ){{\hat{f}}}(\xi )]$. Then, the set $\{K_\lambda |\lambda \in \Lambda \}$ is ${\mathcal {R}}$-bounded on ${\mathcal {L}}(L^q({\mathbb {R}}^d))$ and

$$\begin{aligned} {\mathcal {R}}_{{\mathcal {L}}(L^q({\mathbb {R}}^d))} (\{K_\lambda | \lambda \in \Lambda \}) \le C_{q,d} \max _{|\alpha |\le d+2} C_\alpha . \end{aligned}$$

Lemma 4.1

(Lemma 3.1 in [13]) Let $0<\beta <\pi /2$ and $\nu _0>0$. For any $\lambda \in \Sigma _{\beta ,\nu }$ we have

$$\begin{aligned} |\lambda + |\xi |^2| \ge \sin (\beta /2) (|\lambda | + |\xi |^2). \end{aligned}$$

Proposition 4.1

(Proposition 2.13 in [13]) Let $D\subset {\mathbb {R}}^d$ be a domain and let $\Lambda$ be a domain in ${\mathbb {C}}$. Let $m(\lambda )$ be a bounded function on $\Lambda$ and let $M_m(\lambda ):L^q(D)\mapsto L^q(D)$ be defined as $M_m(\lambda ) f = m(\lambda )f$. Then

$$\begin{aligned} {\mathcal {R}}_{{\mathcal {L}}(L^q(D))} (\{M_m(\lambda )|\lambda \in \Lambda \}) \le C_{d,q,D} \Vert m\Vert _{L^\infty (\Lambda )}. \end{aligned}$$

Proposition 4.2

(Proposition 2.16 in [13] or Proposition 3.4 in [12])

• 1. Let X and Y be Banach spaces and let ${\mathcal {T}}$ and ${\mathcal {S}}$ be ${\mathcal {R}}$-bounded families on ${\mathcal {L}}(X,Y)$. Then ${\mathcal {T}}+{\mathcal {S}} = \{T+S| T\in {\mathcal {T}}, S\in {\mathcal {S}}\}$ is also ${\mathcal {R}}$-bounded on ${\mathcal {L}}(X,Y)$ and

  $$\begin{aligned} {\mathcal {R}}_{{\mathcal {L}}(X,Y)} ({\mathcal {T}} + {\mathcal {S}})\le \mathcal R_{{\mathcal {L}}(X,Y)} ({\mathcal {T}}) + R_{{\mathcal {L}}(X,Y)} ({\mathcal {S}}). \end{aligned}$$

• 2. Let X, Y and Z be Banach spaces and let ${\mathcal {T}}$ and ${\mathcal {S}}$ be ${\mathcal {R}}$-bounded families on ${\mathcal {L}}(X,Y)$ and ${\mathcal {L}}(Y,Z)$ respectively. Then ${\mathcal {S}}{\mathcal {T}} = \{ST| T\in {\mathcal {T}}, S\in {\mathcal {S}}\}$ is ${\mathcal {R}}$-bounded on ${\mathcal {L}}(X,Y)$ and

  $$\begin{aligned} {\mathcal {R}}_{{\mathcal {L}}(X,Z)}({\mathcal {S}}{\mathcal {T}}) \le \mathcal R_{{\mathcal {L}}(X,Y)} ({\mathcal {T}}) {\mathcal {R}}_{\mathcal L(Y,Z)}({\mathcal {S}}). \end{aligned}$$

Remark 4.1

Our plan is to extend the result to the case of a bounded domain including some more general boundary conditions as inflow/outflow or nonhomogeneous Dirichlet boundary conditions.