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A stochastically perturbed fluid-structure interaction problem modeled by a stochastic viscous wave equation. (English) Zbl 1503.35168

The authors derive a stochastic model consisting of a stochastic viscous wave equation written as: \(u_{tt}+2\mu \sqrt{-\Delta}u_{t}-\Delta u=f(u)W(dt,dx)\), and posed in \(\mathbb{R}^{n}\), \(n=1,2\). Here \(W(dt,dx)\) is a noise which accounts for stochastic effects and \(f(u)\) is a nonlinear and Lipschitz continuous function. The initial conditions \(u(0,x)=g(x)\) and \(\partial_{t}u(0,x)=h(x)\) are imposed, \(g\) and \(h\) being continuous versions of functions in \(H^{2}(\mathbb{R}^{n})\). The authors define a mild solution to this problem as a stochastic process \(u(t,x)\) which is jointly measurable and adapted to the filtration \(\mathcal{F}_{t}\) such that \[ u(t,x,\omega)=\int_{\mathbb{R}^{n}}J_{t}(x-y)g(y)dy+\int_{\mathbb{R}^{n}}K_{t}(x-y)h(y)dy+\int_{0}^{t}\int_{\mathbb{R}^{n}}K_{t-s}(x-y)f(u(s,y, \omega))W(dy,ds), \] where \[ J_{t}(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot \xi}e^{-\frac{\left\vert \xi \right\vert}{2}t} \cos(\frac{ \sqrt{3}}{2}\left\vert \xi \right\vert t)+\frac{1}{\sqrt{3}} \sin(\frac{\sqrt{3}}{2}\left\vert \xi \right\vert t)d\xi \] and \[ K_{t}(x)=\frac{1}{(2\pi)^{n}} \int_{\mathbb{R}^{n}}e^{ix\cdot \xi}e^{-\frac{\left\vert \xi \right\vert}{2}t}\frac{\sin (\frac{\sqrt{3}}{2}\left\vert \xi \right\vert t)}{\frac{\sqrt{3}}{2}\left\vert \xi \right\vert}d\xi. \] The first main result of the paper proves the existence of a mild solution to this problem, which is unique up to stochastic modification. For the proof, the authors use Picard iterations, setting the first iterate \(u_{0}\) as: \[ u_{0}(t,x)=\int_{\mathbb{R}^{n}}J_{t}(x-y)g(y)dy+\int_{\mathbb{R}^{n}}K_{t}(x-y)h(y)dy, \] then defining \(u_{k}\) for \(k\geq 1\) as: \[ u_{k}(t,x)=u_{0}(t,x)+\int_{0}^{t}\int_{\mathbb{R}^{n}}K_{t-s}(x-y)f(u_{k-1}(s,y))W(dy,ds). \] They prove that this Picard iteration procedure is well defined at each step and they estimate the difference between consecutive iterates using properties of the kernels. They prove the convergence of the Picard iteration in \(L^{2}(\Omega)\) to some \(u(t,x)\) for each \(t\geq 0\) and \(x\in \mathbb{R}^{n}\). They finally prove that this limit \(u\) satisfies the preceding equation. For the uniqueness, the authors apply Gronwall’s inequality. In the last part of their paper, the authors prove different Hölder and Kolmogorov continuity results. for this mild solution They especially prove that, up to a modification, the stochastic mild solution is \(\alpha \)-Hölder continuous for almost every realization of the solution sample path, where \(\alpha \in [0,1)\) if \(n=1\), and \(\alpha \in [0,1/2)\) if \(n=2\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
76D07 Stokes and related (Oseen, etc.) flows
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74K15 Membranes
74B10 Linear elasticity with initial stresses
60H40 White noise theory
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B20 Perturbations in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35R60 PDEs with randomness, stochastic partial differential equations
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References:

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