Abstract: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.

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中文翻译：

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描述流固耦合的粘性非线性波动方程的确定性不适定性和概率适定性

摘要：我们研究了由 Dirichlet-Neumann 算子描述的粘性耗散效应增强的 $\mathbb {R}^2$ 中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固耦合中，其中 Dirichlet-Neumann 算子对粘性不可压缩流体和弹性结构之间的耦合进行建模。我们表明，尽管粘性正则化，初始数据 $(u,u_t)$ 在 $H^s(\mathbb {R}^2)\times H^{s-1}(\mathbb {R} ^2)$ 是不适定的，只要 $0 < s < s_{cr}$，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 $s_{cr} = 1/2$，这与相关非线性的临界指数相同没有粘性项的波动方程。然后我们证明，如果使用维纳随机化对初始数据进行扰动，这会扰乱频率空间中的初始数据，那么五次非线性粘性波动方程的柯西问题对于超临界指数 $s$ 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出现的非线性粘性波动方程不适定性和概率适定性的结果。

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