Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction
Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction
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DOI：
10.1090/tran/8423
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发表时间：
2021-04
影响因子：
1.3
通讯作者：
Jeffrey Kuan;S. Čanić
中科院分区：
文献类型：
--
作者：
Jeffrey Kuan;S. Čanić
本文研究了$\mathbb{R}^2$中由Dirichlet-Neumann算子描述的粘性耗散效应增强的非线性波动方程的低正则性行为。这种类型的问题出现在流体-结构相互作用中，其中狄利克雷-诺伊曼算子模拟粘性不可压缩流体与弹性结构之间的耦合。我们证明了尽管有粘性正则化，初始数据$(u,u_t)$在$H^s(\mathbb{R}^2)\乘以H^{s-1}(\mathbb{R}^2)$中的柯西问题，在${0 < s < s_{cr}}}$时是不适定的，其中临界指数$s_{cr}$取决于非线性程度。特别地，对于五次非线性$u^5$, $\mathbb{R}^2$中的临界指数为$s_{cr} = 1/2$，这与不含粘性项的相关非线性波动方程的临界指数相同。然后，我们证明了如果初始数据使用Wiener随机化进行扰动，该随机化在频率空间中扰动初始数据，那么对于超临界指数$s$，五次非线性粘性波动方程的Cauchy问题几乎肯定是适定的，使得$-1/6 < s \le s_{cr} = 1/2$。据我们所知，这是第一个显示流固耦合中非线性粘性波动方程的不适定性和概率适定性的结果。
We study low regularity behavior of the nonlinear wave equation in $\mathbb{R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb{R}^2)\times H^{s-1}(\mathbb{R}^2)$, is ill-posed whenever ${{0 < s < s_{cr}}}$, where the critical exponent $s_{cr}$ depends on the degree of nonlinearity. In particular, for the quintic nonlinearity $u^5$, the critical exponent in $\mathbb{R}^2$ is $s_{cr} = 1/2$, which is the same as the critical exponent for the associated nonlinear wave equation without the viscous term. We then show that if the initial data is perturbed using a Wiener randomization, which perturbs initial data in the frequency space, then the Cauchy problem for the quintic nonlinear viscous wave equation is well-posed almost surely for the supercritical exponents $s$ such that $-1/6 < s \le s_{cr} = 1/2$. To the best of our knowledge, this is the first result showing ill-posedness and probabilistic well-posedness for the nonlinear viscous wave equation arising in fluid-structure interaction.