Mathematical and Numerical Analysis of Some FSI Problems

Grandmont, Céline; Lukáčová-Medviďová, Mária; Nečasová, Šárka

doi:10.1007/978-3-0348-0822-4_1

Cited by 13 publications

(7 citation statements)

References 128 publications

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“…We will refer to equation (1) as the viscous nonlinear wave equation (vNLW). We are interested in the Cauchy problem for equation (1), where p > 0 is an odd integer, and µ > 0, supplemented with initial data: u(0, •) = f and u t (0, •) = g, (17) where (f, g) ∈ H s (R 2 ) = H s (R 2 ) × H s−1 (R 2 ). Here H s denotes the usual (inhomogeneous) Sobolev space.…”

Section: Introductionmentioning

confidence: 99%

“…The analysis of fluid-structure interaction problems involving incompressible, viscous fluids and elastic structures started in the early 2000's with works in which the coupling between the fluid and structure was assumed across a fixed fluid-structure interface (linear coupling) as in [1,2,14,22], and was then extended to problems with nonlinear coupling in the works [3,[8][9][10]12,13,[15][16][17][18][19]21,23,24,[29][30][31][32][33][34]. In all these studies, a major underlying reason for the well-posedness is the regularization by the fluid viscosity and the dispersive nature of the wave-like operators in more than one spatial dimension.…”

Section: Introductionmentioning

confidence: 99%

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

Kuan

Čanić²

2021

Trans. Amer. Math. Soc.

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We study low regularity behavior of the nonlinear wave equation in 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 (R 2 ) × H s−1 (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 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 ≤ 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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kuan

Čanić²

2021

Trans. Amer. Math. Soc.

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“…In particular, fluid-structure interaction problems with linear coupling, which is considered in the current work, have been investigated in, e.g., [1,2,15,27]. The more general case of nonlinear coupling, which has been studied in [3,[5][6][7]9,10,[16][17][18]20,21,26,28,29,[31][32][33][34][35][36], allows the fluid domain to change as a function of time, and the coupling conditions between the fluid and structure are evaluated at the current location of the interface, not known a priori. This creates additional (geometric) nonlinearities and generates additional mathematical difficulties.…”

Section: Introductionmentioning

confidence: 99%

“…The resulting behavior is "in between" the wave and heat equations. The viscous wave equation (18) turns out to have just the right scaling and dissipation to allow function-valued mild solutions even in spatial dimension two for the white noise perturbed equation…”

Section: Introductionmentioning

confidence: 99%

A stochastically perturbed fluid-structure interaction problem modeled by a stochastic viscous wave equation

Kuan¹,

Čanić²

2021

Preprint

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We study well-posedness for fluid-structure interaction driven by stochastic forcing. This is of particular interest in real-life applications where forcing and/or data have a strong stochastic component. The prototype model studied here is a stochastic viscous wave equation, which arises in modeling the interaction between Stokes flow and an elastic membrane. To account for stochastic perturbations, the viscous wave equation is perturbed by spacetime white noise scaled by a nonlinear Lipschitz function, which depends on the solution. We prove the existence of a unique function-valued stochastic mild solution to the corresponding Cauchy problem in spatial dimensions one and two. Additionally, we show that up to a modification, the stochastic mild solution is α-Hölder continuous for almost every realization of the solution's sample path, where α ∈ [0, 1) for spatial dimension n = 1, and α ∈ [0, 1/2) for spatial dimension n = 2. This result contrasts the known results for the heat and wave equations perturbed by spacetime white noise, including the damped wave equation perturbed by spacetime white noise, for which a function-valued mild solution exists only in spatial dimension one and not higher. Our results show that dissipation due to fluid viscosity, which is in the form of the Dirichlet-to-Neumann operator applied to the time derivative of the membrane displacement, sufficiently regularizes the roughness of white noise in the stochastic viscous wave equation to allow the stochastic mild solution to exist even in dimension two, which is the physical dimension of the problem. To the best of our knowledge, this is the first result on well-posedness for a stochastically perturbed fluid-structure interaction problem.

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“…Thus, one must solve a moving boundary problem, which introduces additional geometric nonlinearities into the problem. See for example, [3,42,29,14,27,47,48,41,21,22,37,16,15,32,62,33,51,50,49,28,67].…”

mentioning

confidence: 99%

Probabilistic global well-posedness for a viscous nonlinear wave equation modeling fluid-structure interaction

Kuan¹,

Oh²,

Čanić³

2021

Preprint

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We continue the study of low regularity behavior of the viscous nonlinear wave equation (vNLW) on R 2 , initiated by Čanić and the first author (2021). In this paper, we focus on the defocusing quintic nonlinearity and, by combining a parabolic smoothing with a probabilistic energy estimate, we prove almost sure global well-posedness of vNLW for initial data in H s (R 2 ), s > − 1 5 , under a suitable randomization.

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Mathematical and Numerical Analysis of Some FSI Problems

Cited by 13 publications

References 128 publications

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

A stochastically perturbed fluid-structure interaction problem modeled by a stochastic viscous wave equation

Probabilistic global well-posedness for a viscous nonlinear wave equation modeling fluid-structure interaction

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