Pointwise space–time estimates of non-isentropic compressible micropolar fluids

Abstract

For the non-isentropic compressible micropolar fluids in three dimensions, we show that the space–time behaviors of the fluid density, momentum and energy contain both generalized Huygens’ wave and diffusion wave as the non-isentropic compressible Navier–Stokes equation, while it only contains the diffusion wave for the micro-rational velocity. All of the previous estimates containing Huygens’ waves for wave behaviors of compressible flow models rely heavily on the conservative structure, for instance, the isentropic and non-isentropic Navier–Stokes equations. In this paper, after using the decomposition of fluid and electromagnetic parts to study three smaller Green’s matrices, we find that one of them is like a non-isentropic Navier–Stokes equation, but its energy is not conservative anymore due to the presence of the micro-rational velocity. We overcome this difficulty by providing a refined estimate for this Green’s matrix. Another difficulty is in proving the pointwise estimate of the micro-rational velocity only contains the diffusion wave, although its nonlinear terms contain both the Huygens’ wave and the diffusion wave. It is solved by using the relation of these two waves and developing new nonlinear estimates. Our pointwise estimate directly yields $L^p$-estimate with $p>1$, which is a generalization of the usual $L^2$-estimate.

Appendix

      The first lemma is used to derive H-wave for the Green’s function $G^1(x,t)$ in Sect. 2.

Lemma 3.1

[25, 26] Let $\mathbf {w}(x, t)$ be the inverse Fourier transform of $\sin c|\xi |t/(c|\xi |)$, then

$$\begin{aligned} \begin{array}{rl} |\mathbf {w}*(1+t)^{-\frac{l}{2}}\mathrm{e}^{-\frac{|x|^2}{1+t}}| &{} \le C(1+t)^ {-\frac{l}{2}}\mathrm{e}^{-\frac{(|x|-ct)^2}{1+t}},\\ |\mathbf {w}_t*(1+t)^{-\frac{l}{2}}\mathrm{e}^{-\frac{|x|^2}{1+t}}|&{} \le C(1+t)^ {-\frac{l+2}{2}}\mathrm{e}^{-\frac{(|x|-ct)^2}{1+t}},\ l\ge 0. \end{array} \end{aligned}$$

We also need the following lemma when describing the singular part of the short wave:

Lemma 3.2

(Wang and Yang [36]) If $\mathrm{supp}\hat{f}(\xi )\subset O_K=:{\{\xi , |\xi |\ge K>0\}}$, and $\hat{f}(\xi )$ satisfies

$$\begin{aligned} |D_\xi ^\beta \hat{f}(\xi )|\le C|\xi |^{-|\beta |-1}, \end{aligned}$$

then there exist distributions $f_1(x), f_2(x)$ and a constant $C_0$ such that $f(x)=f_1(x)+f_{2}(x)+C_{0}\delta (x)$, where $\delta (x)$ is the Dirac function. Furthermore, for any $|\alpha |\ge 0$ and any positive integer N, we have

$$\begin{aligned} |D_x^\alpha f_1(x)|\le C(1+|x|^2)^{-N},\ \Vert f_{2}\Vert _{L^1}\le C,\ \mathrm{supp}f_{2}(x)\subset \{x;|x|<\eta _0\ll 1\}. \end{aligned}$$

The last two lemmas are used for initial propagation and nonlinear coupling, respectively. We just state several typical cases here.

Lemma 3.3

(Wu and Wang [39]) There exists a constant $C>0$ such that:

$$\begin{aligned} \begin{array}{ll} \displaystyle \int \limits _{\mathbb R^3}\mathrm{e}^{-\frac{|x-y|^2}{C(1+t)}}\big (1+|y|^2\big )^{-r}\mathrm{d}y \le C\Big (1+\frac{|x|^2}{1+t}\Big )^{-r},\ \ \mathrm{for}\ r>\frac{3}{2},\\ \displaystyle \int \limits _{\mathbb R^3}\mathrm{e}^{-\frac{(|x-y|-ct)^2}{C(1+t)}}\big (1+|y|^2\big )^ {-r_1}\mathrm{d}y\le C\Big (1+\frac{(|x|-ct)^2}{1+t}\Big )^{-\frac{3}{2}},\ \ \mathrm{for}\ r_1>\frac{21}{10}. \end{array} \end{aligned}$$

(3.25)

Lemma 3.4

(Liu and Noh [25]) There exists a constant $C>0$ such that

$$\begin{aligned} \begin{array}{ll} \displaystyle K_1=\int \limits _0^t\int \limits _{\mathbb {R}^3}(1+t-s)^{-2}\Big (1+\frac{|x-y|^2}{1+t-s}\Big )^ {-2}(1+s)^{-3}\Big (1+\frac{|y|^2}{1+s}\Big )^{-3}\mathrm{d}y\mathrm{d}s\\ \ \ \ \ \le C(1+t)^{-2}\big (1+\frac{|x|^2}{1+t}\big )^{-\frac{3}{2}},\\ \displaystyle K_2=\int \limits _0^t\int \limits _{\mathbb {R}^3}(1+t-s)^{-2}\Big (1+\frac{|x-y|^2}{1+t-s}\Big )^ {-2}(1+s)^{-4}\Big (1+\frac{(|y|-cs)^2}{1+s}\Big )^{-3}\mathrm{d}y\mathrm{d}s\\ \ \ \ \ \le C(1+t)^{-2}\Big (\big (1+\frac{|x|^2}{1+t}\big )^{-\frac{3}{2}}+\big (1+ \frac{(|x|-ct)^2}{1+t}\big )^{-\frac{3}{2}}\Big ),\\ K_3=\displaystyle \int \limits _{0}^{t}\int \limits _{\mathbb {R}^3}(1+t-s)^{-\frac{5}{2}}\mathrm{e}^{- \frac{(|x-y|-c(t-s))^2}{C(1+t-s)}}(1+s)^{-4}\Big (1+\frac{(|y|-cs)^2}{1+s}\Big )^{-3}\mathrm{d}y\mathrm{d}s\\ \ \ \ \ \le C(1+t)^{-2}\Big (\big (1+\frac{|x|^2}{1+t}\big )^{-\frac{3}{2}}+\big (1+ \frac{(|x|-ct)^2}{1+t}\big )^{-\frac{3}{2}}\Big ). \end{array} \end{aligned}$$

The estimate $K_3$ is key to derive the generalized Huygens’ principle for the previous results on the conservative system such as the isentropic Navier–Stokes equations mentioned in the Introduction, since the extra $(1+t)^{-\frac{1}{2}}$ decay rate in the former term of the convolution (3.25) is from the conservative structure. In our situation, the energy is non-conservative; we have to find out this extra decay rate from the Green’s function. Fortunately, we get it and then realize that the model (1.1) is naturally perfect from the perspective of deducing the generalized Huygens’ principle here.