Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction

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.