论文标题
关于$ \ mathbb r^3 $的通用压缩两流体模型的不稳定性
On instability of a generic compressible two-fluid model in $\mathbb R^3$
论文作者
论文摘要
我们关注整个空间中通用可压缩的两流体模型的不稳定性，其中毛细管压力$ f（α^ - ρ^ - ）= p^+ - p^ - \ neq 0 $。对于毛细管压力是在平衡附近的严格降低功能的情况，即$ f'（1）<0 $，在某些较小度假设下，Evje-wang-wen建立了三维Cauchy问题的恒定平衡状态的全局稳定性。最近，Wu-Yao-Zhang证明了case $ p^+= p^ - $的恒定平衡状态的全局稳定性（对应于$ f'（1）= 0 $）。在这项工作中，我们研究了恒定平衡状态的不稳定性，因为毛细管压力是在平衡附近的严格增加功能，即$ f'（1）> 0 $。首先，通过采用Hodge分解技术并对相应的线性化系统的绿色功能进行详细分析，我们构建了线性化问题的解决方案，这些解决方案在Sobolev空间$ H^K $中呈指数增长，从而导致了线性化问题的全球不稳定结果。此外，借助全球线性不稳定性结果和原始非线性系统的经典解决方案的局部存在定理，我们可以通过对Hadamard的意义上对非线性问题的不稳定性，通过对半群的特性进行微妙的分析。因此，我们的结果表明，对于$ f'（1）> 0 $，两流体模型的持续平衡状态在全球范围内是线性的，并且从哈达姆的意义上讲是不稳定的，并且在hadamard的意义上是不稳定的，这与$ $ f'（1）<0 $和$ p^+= p^+= p^ - $ f- $ f- $ f- $ f- $ f'（1）= 0 $ f'（1）= 0 $ f'（1）模型在全球范围内是非线性的。
We are concerned with the instability of a generic compressible two-fluid model in the whole space $\mathbb{R}^3$, where the capillary pressure $f(α^-ρ^-)=P^+-P^-\neq 0$ is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, $f'(1)<0$, Evje-Wang-Wen established global stability of the constant equilibrium state for the three-dimensional Cauchy problem under some smallness assumptions. Recently, Wu-Yao-Zhang proved global stability of the constant equilibrium state for the case $P^+=P^-$ (corresponding to $f'(1)=0$). In this work, we investigate the instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, $f'(1)>0$. First, by employing Hodge decomposition technique and making detailed analysis of the Green's function for the corresponding linearized system, we construct solutions of the linearized problem that grow exponentially in time in the Sobolev space $H^k$, thus leading to a global instability result for the linearized problem. Moreover, with the help of the global linear instability result and a local existence theorem of classical solutions to the original nonlinear system, we can then show the instability of the nonlinear problem in the sense of Hadamard by making a delicate analysis on the properties of the semigroup. Therefore, our result shows that for the case $f'(1)>0$, the constant equilibrium state of the two-fluid model is linearly globally unstable and nonlinearly locally unstable in the sense of Hadamard, which is in contrast to the cases $f'(1)<0$ and $P^+=P^-$ (corresponding to $f'(1)=0$) where the constant equilibrium state of the two--fluid model is nonlinearly globally stable.
