AI 中文总结
本文针对具有弱衰减初值的三维Chaplygin气体无旋可压缩欧拉方程，在给定的初值条件下，通过建立相关估计证明了其全局经典解的存在性。
AI 中文摘要
我们研究三维可压缩等熵Chaplygin气体欧拉方程的全局经典解问题，其方程组为：\\[ \begin{cases} \partial_t\rho + \mathrm{div}(\rho v) = 0,\\\\ \partial_t(\rho v) + \mathrm{div}(\rho v \otimes v) + \nabla p = 0,\\\\ \rho(0,x) = \bar\rho + \varepsilon\rho_0(x),\\ v(0,x) = \varepsilon v_0(x). \end{cases} \\] 其中$\bar\rho>0$为常数，$\varepsilon>0$为小量，状态方程为$p=p(\rho)=P_0-\frac{D}{\rho}$，$P_0$和$D$为正常数。三维Chaplygin气体可压缩欧拉方程是具有完全线性退化特征值的多维非线性对称双曲系统的原型，A. Majda提出了一个基本猜想：当$\rho_0, v_0\in H^s(\mathbb{R}^3)$且$s>\frac{5}{2}$时，除非$\rho, v$在有限时间内爆破，否则其通常存在全局经典解$(\rho, v)$满足$(\rho-\bar\rho, v)\in C([0,\infty), H^s(\mathbb{R}^3)) \cap C^1([0,\infty), H^{s-1}(\mathbb{R}^3))$。本文在以下假设下：对任意固定常数$\mu$满足$0<\mu<1/2$、整数$N\geq 15$，$\mathrm{rot}\\,v_0(x) \equiv 0$，且$\\|(\rho_0, v_0)\\|_{H^{N}(\mathbb{R}^3)}+\sum_{|a|\leq 13} \\|\langle x\rangle^{1+\mu} \nabla^a(\rho_0,v_0)\\|_{L^2(\mathbb{R}^3)} \leq 1$，证明了经典解$(\rho, v)$全局存在。本文的主要工作包括：建立三维Chaplygin气体势流方程的一系列新的能量界衰减估计、加权逐点时空$L^\infty$-$L^2$估计以及加权Strichartz型估计。
英文摘要
We are concerned with the global classical solution problem of 3D compressible isentropic Euler equations of Chaplygin gases \[ \begin{cases} \partial_tρ+ \mathrm{div}(ρv) = 0,\\ \partial_t(ρv) + \mathrm{div}(ρv \otimes v) + \nabla p = 0,\\ ρ(0,x) = \barρ+ \varepsilonρ_0(x),\ v(0,x) = \varepsilon v_0(x). \end{cases} \] where $\barρ>0$ is a constant, $\varepsilon>0$ is small, the state equation is $p=p(ρ)=P_0-\frac{D}ρ$ with $P_0$ and $D$ being some positive constants. For the 3D compressible Euler equations of Chaplygin gases, which are a prototype of multidimensional nonlinear symmetric hyperbolic systems with totally linearly degenerate eigenvalues, there is a basic conjecture imposed by A. Majda: it typically has a global classical solution $(ρ, v)$ with $(ρ-\barρ, v)\in C([0,\infty), H^s(\Bbb R^3)) \cap C^1([0,\infty), H^{s-1}(\Bbb R^3))$ when $(ρ_0, v_0)\in H^s(\Bbb R^3)$ with $s>\frac52$ unless $(ρ, v)$ itself blows up in finite time. In this paper, under the assumptions that for any fixed constant $μ$ with $0<μ<1/2$, integer $N\geq 15$, $\mathrm{rot}\,v_0(x) \equiv 0$ and \[ \|(ρ_0, v_0)\|_{H^{N}(\mathbb{R}^3)}+\sum_{|a|\leq 13} \|\langle x\rangle^{1+μ} \nabla^a(ρ_0,v_0)\|_{L^2(\mathbb{R}^3)} \leq 1, \] we show that the classical solution $(ρ, v)$ exists globally. Our main ingredients include: establishing a series of new decay estimates of energy bounds, weighted pointwise space-time $L^\infty$-$L^2$ estimates and weighted Strichartz-type estimates for the 3D potential flow equation of Chaplygin gases.
