In this article we study the boundary layers for the subcritical modes of the viscous Linearized Primitive Equations (LPEs) in a cube at small viscosity. The boundary layers include the parabolic boundary layers, ordinary boundary layers, and their interaction-corner layers. The boundary layer correctors are determined by a phenomenological study reminiscent of the Prandtl corrector approach and then a rigorous convergence result is proved which a posteriori justifies the phenomenological study.

The aim of this article is to show the global existence of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general Lévy noise. The classical approach based on using directly the Galerkin approximation is not valid. Instead, our approach is based on using appropriate approximations for the monotone operator, Galerkin approximations and on the theory of martingale solutions.

In this article we study approximations of monotone operators acting in Banach spaces by nonlinear operators acting in Hilbert spaces. The motivation behind this study appears in the stochastic context. We give a better existence result, than previously known, for the Ladyzenskaya equations (Smagorinsky model of turbulence).

We give an asymptotic upper bound for the kth twisted eigenvalue of the linearized Allen–Cahn operator in terms of the kth eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic limit of the zero sets of the Allen–Cahn critical points. We use an argument based on the notion of second inner variation developed in Le (On the second inner variations of Allen–Cahn type energies and applications to local minimizers. J. Math. Pures Appl. (9) 103, no. 6, (2015), 1317– 1345).

In this article we study an FSI problem that models the mechanical interaction between blood flow and a thrombus with Hookean elasticity, using a phase-field method to describe a two-phase system, with the interface between two different phases given by a thin smooth transition layer. This model is studied in a bounded domain in dimensions d=3. We prove the existence of a unique local solution.

We provide existence and upper bounds for the time-averaged energy dissipation rate for the martingale solutions of the incompressible 3D NSE in a cube subject to a shear induced by a square-integrable Lévy noise on one wall by constructing an appropriate stochastic background flow that depends on the stochastic forcing.

We prove the existence of martingale solutions to a stochastic fluid-structure interaction problem involving a viscous, incompressible fluid flow, modeled by the Navier-Stokes equations, through a deformable elastic tube modeled by shell equations. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic coupling conditions at the fluid-structure interface. This article considers the case where the structure can have unrestricted displacement and explores the Navier-slip boundary condition imposed at the fluid-structure interface, displacement of which is not known a priori and is itself a part of the solution. The proof takes a constructive approach based on a Lie splitting scheme. The geometric nonlinearity stemming from the nonlinear coupling, the possibility of random fluid domain degeneracy, the potential jumps in the tangential components of the fluid and structure velocities at the moving interface and the low regularity of the structure velocity require the development of new techniques that lead to the existence of martingale solutions.

This paper presents a constructive method to investigate solutions of a nonlinearly coupled stochastic fluid-structure interaction (FSI) problem. The focus is on a stochastically forced benchmark problem involving a linearly elastic membrane/shell that interacts with a two dimensional flow of a viscous, incompressible Newtonian fluid across a moving interface. This benchmark problem incorporates the main mathematical difficulties associated with nonlinearly coupled stochastic FSI problems. The fluid is modeled by the 2D Navier-Stokes equations, while the membrane is modeled by the linearly elastic membrane/shell equations. The fluid and the structure are coupled across the moving interface via a two-way coupling that ensures continuity of velocities and continuity of contact forces at the fluid-structure interface. The stochastic noise is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The noise is multiplicative depending on the structure displacement, structure velocity and the fluid velocity. The main result of this manuscript is a constructive proof of the existence of weak martingale solutions to this nonlinear stochastic fluid-structure interaction problem. Namely, we prove the existence of solutions that are weak in the analytical sense and in the probabilistic sense. In other words, we show that despite the roughness, the underlying nonlinear deterministic fluid-structure interaction problem is robust to noise. To the best of our knowledge, this is the first result in the field of stochastic PDEs that addresses the question of existence of solutions on moving domains involving incompressible fluids, where the displacement of the boundary and the motion of the fluid domain are random variables that are not known a priori and are parts of the solution itself.

In this paper we study a nonlinear stochastic fluid-structure interaction problem with a multiplicative, white-in-time noise. The problem consists of the Navier-Stokes equations describing the flow of an incompressible, viscous fluid in a 2D cylinder interacting with an elastic lateral wall whose elastodynamics is described by a membrane/shell equation. The stochastic noise is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic conditions assumed at the moving interface, which is a random variable not known a priori. In particular, we consider the case where the structure is allowed to have non-zero longitudinal displacement.

We study the existence of weak martingale solutions to a stochastic moving boundary problem arising from the interaction between an isentropic compressible fluid and a viscoelastic structure. In the model, we consider a three-dimensional compressible isentropic fluid with adiabatic constant > 3/2 interacting dynamically with an elastic structure on the boundary of the fluid domain described by a plate equation, under the additional influence of stochastic perturbations which randomly force both the compressible fluid and elastic structure equations in time. The problem is nonlinearly coupled in the sense that the a priori unknown (and random) displacement of the elastic structure from its reference configuration determines the a priori unknown (and random) time-dependent fluid domain on which the compressible isentropic Navier-Stokes equations are posed. We use a splitting method, consisting of a fluid and structure subproblem, to construct random approximate solutions to an approximate Galerkin form of the problem with artificial viscosity and artificial pressure. We introduce stopped processes of structure displacements which handle the issues associated with potential fluid domain degeneracy. In this splitting scheme, we handle mathematical difficulties associated with the a priori unknown and time-dependent fluid domain by using an extension of the fluid equations to a fixed maximal domain and we handle difficulties associated with imposing the no-slip condition in the stochastic setting by using a novel penalty term defined on an external tubular neighborhood of the moving fluid-structure interface. To the best of our knowledge, this is the first well-posedness result for stochastic fluid-structure interaction with compressible fluids.

In this paper, we investigate a nonlinear fluid-structure interaction (FSI) problem involving the Navier-Stokes equations, which describe the flow of an incompressible, viscous fluid within a 3D domain interacting with a viscoelastic lateral wall. The elastodynamics of the wall are modeled by a two-dimensional plate equation with fractional damping, accounting for displacement in all three directions. The system is nonlinearly coupled through kinematic and dynamic conditions imposed at the time-varying fluid- structure interface. We establish a hidden spatial regularity result for the structure, enabling us to show that there is a finite time interval within which the structure self-contact will not occur. Additionally, we demonstrate temporal regularity for both the structure and fluid velocities, forming the basis for a new compactness result in the context of three-dimensional structural displacement. Leveraging these regularity results, we prove the existence of a local-in-time weak solution for the FSI problem. A constructive proof is provided using time discretization via the Lie operator splitting method.