Stochastic compressible Euler equations and inviscid limits

We prove the existence of a unique local strong solution to the stochastic compressible Euler system with nonlinear multiplicative noise. This solution exists up to a positive stopping time and is strong in both the PDE and probabilistic sense. Based on this existence result, we study the inviscid limit of the stochastic compressible Navier--Stokes system. As the viscosity tends to zero, any sequence of finite energy weak martingale solutions converges to the compressible Euler system.Comment: 26 page

Weak and strong solutions for polymeric fluid-structure interaction of Oldroyd-B type

We prove the existence of weak solutions and a unique strong solution to the Oldroyd-B dumbbell model describing the evolution of a two-dimensional dilute polymer fluid interacting with a one-dimensional viscoelastic shell. The polymer fluid consists of a mixture of an incompressible viscous solvent and a solute comprising two massless beads connected with Hookean springs. This solute-solvent mixture then interacts with a flexible structure that evolves in time. An arbitrary nondegenerate reference domain for the polymer fluid is allowed and both solutions exist globally in time provided no self-intersection of the structure occurs.Comment: 29 page

An incompressible polymer fluid interacting with a Koiter shell

We study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier-Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker-Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.Comment: 38 page

The stochastic compressible Navier-Stokes system on the whole space and some singular limits

Firstly, we show the existence of at least one non-trivial solution to the stochastically forced compressible Navier–Stokes system defined on the whole Euclidean space. This solution is deterministically weak in the usual sense of distributions but also weak in the sense of probability, the latter meaning that the underlying probability space, as well as the stochastic driving force, are also unknowns. Secondly, we study various asymptotic results for the above mentioned system when the microscopic time and space variables are rescaled appropriately. Different rescaling leads to various singular versions of this system with coefficients which either blow up or dissipate when they are made small. Subsequently, we are able to show that any family of the solutions constructed above parametrised by the singular coefficients converges to solutions of other fluid dynamic models like the incompressible Navier–Stokes system and the compressible Euler system with corresponding stochastic forcing terms. Crucially, we also consider the case when rotation in the fluid is taken into account

Ladyzhenskaya-Prodi-Serrin condition for fluid-structure interaction systems

We consider the interaction of a viscous incompressible fluid with a flexible shell in three space dimensions. The fluid is described by the three-dimensional incompressible Navier--Stokes equations in a domain that is changing in accordance with the motion of the structure. The displacement of the latter evolves along a visco-elastic shell equation. Both are coupled through kinematic boundary conditions and the balance of forces. We prove a counterpart of the classical Ladyzhenskaya-Prodi-Serrin condition yielding conditional regularity and uniqueness of a solution. Our result is a consequence of the following three ingredients which might be of independent interest: {\bf (i)} the existence of local strong solutions, {\bf (ii)} an acceleration estimate (under the Serrin assumption) ultimately controlling the second-order energy norm, and {\bf (iii)} a weak-strong uniqueness theorem. The first point, and to some extent, the last point were previously known for the case of elastic plates, which means that the relaxed state is flat. We extend these results to the case of visco-elastic shells, which means that more general reference geometries are considered such as cylinders or spheres. The second point, i.e. the acceleration estimate for three-dimensional fluids is new even in the case of plates.Comment: 42 page

Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model

This paper investigates the mathematical properties of a stochastic version of the balanced 2D thermal quasigeostrophic (TQG) model of potential vorticity dynamics. This stochastic TQG model is intended as a basis for parametrisation of the dynamical creation of unresolved degrees of freedom in computational simulations of upper ocean dynamics when horizontal buoyancy gradients and bathymetry affect the dynamics, particularly at the submesoscale (250m-10km). Specifically, we have chosen the SALT (Stochastic Advection by Lie Transport) algorithm introduced in [25] and applied in [11,12] as our modelling approach. The SALT approach preserves the Kelvin circulation theorem and an infinite family of integral conservation laws for TQG. The goal of the SALT algorithm is to quantify the uncertainty in the process of up-scaling, or coarse-graining of either observed or synthetic data at fine scales, for use in computational simulations at coarser scales. The present work provides a rigorous mathematical analysis of the solution properties of the thermal quasigeostrophic (TQG) equations with stochastic advection by Lie transport (SALT) [27,28].Comment: 38 page

Space-time approximation of parabolic systems with variable growth

We study a parabolic system with $p(t,x)$-structure under Dirichlet boundary conditions. In particular, we deduce the optimal convergence rate for the error of the gradient of a finite element based space-time approximation. The error is measured in the quasi norm and the result holds if the exponent $p(t,x)$ is $(\alpha_t, \alpha_x)$-H\"{o}lder continuous.Comment: 36 page