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We consider the interaction of a compressible fluid with a flexible plate in two space dimensions. The fluid is described by the Navier--Stokes equations in a domain that is changing in accordance with the motion of the structure. The displacement of the latter evolves according to a beam equation. Both are coupled through kinematic boundary conditions and the balance of forces. We prove that for any weak solution to the coupled system, which satisfies certain additional regularity requirements, no contact occurs between the elastic wall and the bottom of the fluid cavity. This applies to both isentropic and heat-conducting fluids. As a special case of our general theory we extend the unconditional result from Grandmont and Hillairet (Arch. Ration. Mech. Anal. 220, 1283--1333, 2016) on incompressible fluids from visco-elastic to perfectly elastic plates.

We consider a viscous incompressible fluid interacting with a linearly elastic shell of Koiter type which is located at some part of the boundary. Recently models with stochastic perturbation in the shell equation have been proposed in the literature but only analysed in simplified cases. We investigate the full model with transport noise, where (a part of) the boundary of the fluid domain is randomly moving in time. We prove the existence of a weak martingale solution to the underlying system., Comment: 44 pages. To appear in Journal of Nonlinear Science

We construct a unique local strong solution to the finitely extensible nonlinear elastic (FENE) dumbbell model of Warner-type for an incompressible polymer fluid (described by the Navier-Stokes-Fokker-Planck equations) interacting with a flexible elastic shell. The latter occupies the flexible boundary of the polymer fluid domain and is modeled by a beam equation coupled through kinematic boundary conditions and the balance of forces. In the 2D case for the co-rotational Fokker-Planck model we obtain global-in-time strong solutions. A main step in our approach is the proof of local well-posedness for just the solvent-structure system in higher-order topologies which is of independent interest. Different from most of the previous results in the literature, the reference spatial domain is an arbitrary smooth subset of $\mathbb{R}^3$, rather than a flat one. That is, we cover viscoelastic shells rather than elastic plates. Our result also supplements the existing literature on the Navier-Stokes-Fokker-Planck equations posed on a fixed bounded domain., Comment: We added the proof of existence of a global strong solution for the co-rotational model in 2D

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 pages

We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative It\^o-type noise., Comment: We added extensive numerical simulations. They underline the sharpness of our theoretical results and hinge sensitively on the compatibility of the data

We consider the incompressible Navier-Stokes equations in a moving domain whose boundary is prescribed by a function $\eta=\eta(t,y)$ (with $y\in\mathbb R^2$) of low regularity. This is motivated by problems from fluid-structure interaction. We prove partial boundary regularity for boundary suitable weak solutions assuming that $\eta$ is continuous in time with values in the fractional Sobolev space $W^{2-1/p,p}_y$ for some $p>15/4$ and we have $\partial_t\eta\in L_t^{3}(W^{1,q_0}_y)$ for some $q_0>2$. The existence of boundary suitable weak solutions is a consequence of a new maximal regularity result for the Stokes equations in moving domains which is of independent interest., Comment: A variant of Theorem 3.1 was previously included in arXiv:2207.14159 but has been removed in the revised version. arXiv admin note: text overlap with arXiv:2208.00415

We consider the steady Stokes equations in a bounded domain with forcing in divergence form supplemented with no-slip boundary conditions. We provide a maximal regularity theory in Campanato spaces (inlcuding $\mathrm{BMO}$ and $C^{0,\alpha}$ for $0<\alpha <1$ as special cases) under minimal assumptions on the regularity of the underlying domain. Our approach is based on pointwise multipliers in Campanto spaces., Comment: To appear in Indiana Univ. Math. J

Embeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. The equivalence of their Gagliardo-Slobodeckij norms to norms defined via Littlewood-Paley decompostions, via oscillations, or via Besov type difference quotients is also established. These equivalences, of independent interest, are a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.

We consider the 3D stochastic Navier-Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates in for the energy error with respect to convergence in probability, that is convergence of order 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution., Comment: We corrected some typos. arXiv admin note: text overlap with arXiv:2109.06495

We prove strong rate resp. weak rate ${\mathcal O}(\tau)$ for a structure preserving temporal discretization (with $\tau$ the step size) of the stochastic Allen-Cahn equation with additive resp. multiplicative colored noise in $d=1,2,3$ dimensions. Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related Kolmogorov equation, and then leads to weak rate ${\mathcal O}(\tau)$ in the presence of multiplicative noise. Hence, we obtain twice the rate of convergence known for the strong error in the presence of multiplicative noise.

We prove partial regularity of suitable weak solutions to the Navier--Stokes equations at the boundary in irregular domains. In particular, we provide a criterion which yields continuity of the velocity field in a boundary point and obtain solutions which are continuous in a.a. boundary boundary point (their existence is a consequence of a new maximal regularity result for the Stokes equations in domains with minimal regularity). We suppose that we have a Lipschitz boundary with locally small Lipschitz constant which belongs to the fractional Sobolev space $W^{2-1/p,p}$ for some $p>\frac{15}{4}$. The same result was previously only known under the much stronger assumption of a $C^2$-boundary., Comment: We added the proof of the maximal regularity theory for the unsteady Stokes system in irregular domains. A variant of it was previously included in arXiv:2207.14159 but has been removed in the revised version. arXiv admin note: text overlap with arXiv:2207.14159

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our main result is the existence of a unique global strong solution. Previously, only the ideal case of a flat reference geometry was considered such that the structure can only move in vertical direction. We allow for a general geometric set-up, were the structure can even occupy the complete boundary. Our main tool -- being of independent interest -- is a maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity. In particular, we can control the velocity in $W^{2,2}$ in terms of a forcing in $L^2$ provided the boundary belongs roughly to $W^{3/2,2}$. This is applied to the momentum equation in the moving domain (for a fixed time) with the material derivative as right-hand side. Since the moving boundary belongs a priori only to the class $W^{2,2}$, known results do not apply here as they require a $C^2$-boundary., Comment: The part on higher order estimates has been removed. The maximal regularity theory for the unsteady Stokes system will appear in the revised version of arXiv:2208.00415

We consider the barotropic Navier--Stokes system driven by a physically well-motivated transport noise in both continuity as well as momentum equation. We focus on three different situations: (i) the noise is smooth in time and the equations are understood as in the sense of the classical weak deterministic theory, (ii) the noise is rough in time and we interpret the equations in the framework of rough paths with unbounded rough drivers and (iii) we have a Brownian noise of Stratonovich type and study the existence of martingale solutions. The first situation serves as an approximation for (ii) and (iii), while (ii) and (iii) are motivated by recent results on the incompressible Navier--Stokes system concerning the physical modeling as well as regularization by noise.

We propose and study a temporal, and spatio-temporal discretisation of the 2D stochastic Navier--Stokes equations in bounded domains supplemented with no-slip boundary conditions. Considering additive noise, we base its construction on the related nonlinear random PDE, which is solved by a transform of the solution of the stochastic Navier--Stokes equations. We show strong rate (up to) $1$ in probability for a corresponding discretisation in space and time (and space-time). Convergence of order (up to) 1 in time was previously only known for linear SPDEs.

We study a finite-element based space-time discretisation for the 2D stochastic Navier-Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space. This was previously only known in the space-periodic case, where higher order energy estimates for any given (deterministic) time are available. In contrast to this, in the Dirichlet-case estimates are only known for a (possibly large) stopping time. We overcome this problem by introducing an approach based on discrete stopping times. This replaces the localised estimates (with respect to the sample space) from earlier contributions., Comment: Statement and proof of Lemma 3.1. (c) have been revised

We consider the physical setup of a three-dimensional fluid-structure interaction problem. A viscous compressible gas or liquid interacts with a nonlinear, visco-elastic, three-dimensional bulk solid. The latter is described by a hyperbolic evolution with a non-convex elastic energy functional. The fluid is modelled by the compressible Navier--Stokes equations with a barotropic pressure law. Due to the motion of the solid, the fluid domain is time-changing. Our main result is the long-time existence of a weak solution to the coupled system until the time of a collision. The nonlinear coupling between the motions of the two different matters is established via the method of minimising movements. The motion of both the solid and the fluid is chosen via an incrimental minimization with respect to dissipative and static potentials. These variational choices together with a careful construction of an underlying flow map for our approximation then directly result in the pressure gradient and the material time derivatives.

We study the full Navier--Stokes--Fourier system governing the motion of a general viscous, heat-conducting, and compressible fluid subject to stochastic perturbation. The system is supplemented with non-homogeneous Neumann boundary conditions for the temperature and hence energetically open. We show that, in contrast with the energetically closed system, there exists a stationary solution. Our approach is based on new global-in-time estimates which rely on the non-homogeneous boundary conditions combined with estimates for the pressure.

We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality., Comment: More realistic pressure law including the term $\varrho\vartheta$

We are concerned with the long time behavior of the stochastic Navier--Stokes system for compressible fluids in dimension two and three. In this setting, the part of the phase space occupied by the solution depends sensitively on the choice of the initial state. Our main results are threefold. (i) The kinetic energy of a solution is universally and asymptotically bounded, independent of the initial datum. (ii) Time shifts of a solution with initially controlled energy are asymptotically compact and generate an entire solution defined for all $t\in R$. (iii) Every solution with initially controlled energy generates a stationary solution and even an ergodic stationary solution on the closure of the convex hull of its $\omega$--limit set.

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 pages

We study the three-dimensional incompressible Euler equations subject to stochastic forcing. We develop a concept of dissipative martingale solutions, where the nonlinear terms are described by generalised Young measures. We construct these solutions as the vanishing viscosity limit of solutions to the corresponding stochastic Navier-Stokes equations. This requires a refined stochastic compactness method incorporating the generalised Young measures. Our solutions satisfy a form of the energy inequality which gives rise to a weak-strong uniqueness result (pathwise and in law). A dissipative martingale solution coincides (pathwise or in law) with the strong solution as soon as the latter exists.

A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangementinvariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.

We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space-time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskii spaces and therefore cover situations when the (gradient of) the solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolution $h$ and $\tau$. The theoretical error analysis is complemented by numerical experiments., Comment: Source file for experiments included in submission

In this paper we consider the steady Baldwin-Lomax model, which is a rotational model proposed to describe turbulent flows at statistical equilibrium. The Baldwin-Lomax model is specifically designed to address the problem of a turbulent motion taking place in a bounded domain, with Dirichlet boundary conditions at solid boundaries. The main features of this model are the degeneracy of the operator at the boundary and a formulation in velocity/vorticity variables. The principal part of the operator is non-linear and it is degenerate, due to the presence (as a coefficient) of a power of the distance from the boundary: This fact makes the existence theory naturally set in the framework of appropriate weighted-Sobolev spaces., Comment: 31 pages

We consider a dilute suspension of dumbbells joined by a finitely extendible nonlinear elastic (FENE) connector evolving under the classical Warner potential $U(s)=-\frac{b}{2} \log(1-\frac{2s}{b})$, $s\in[0,\frac{b}{2})$. The solvent under consideration is modelled by the compressible Navier--Stokes system defined on the torus $\mathbb{T}^d$ with $d=2,3$ coupled with the Fokker--Planck equation (Kolmogorov forward equation) for the probability density function of the dumbbell configuration. We prove the existence of a unique local-in-time solution to the coupled system where this solution is smooth in the spacetime variables and interpreted weakly in the elongation variable. Our result holds true independently of whether or not the centre-of-mass diffusion term is incorporated in the Fokker--Planck equation., Comment: 28 pages, comments are welcomed

We construct a Lipschitz truncation which approximates functions of bounded variation in the area-strict metric. The Lipschitz truncation changes the original function only on a small set similar to Lusin's theorem. Previous results could only give estimates on the Lebesgue measure of the set where the Lipschitz approximations differ from the original function.

We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible oscillations and/or concentrations in the associated generating sequence. Unlike the conventional measure-valued solutions or rather their expected values, the dissipative solutions comply with a natural compatibility condition - they are classical solutions as long as they enjoy certain degree of smoothness.

We study stochastic Navier-Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measure in the $L^\infty_tL^2_x\cap L^2_tW^{1,2}_x$-norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from [E. Carelli, A. Prohl: Rates of convergence for discretizations of the stochastic incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 50(5), 2467-2496. (2012)] where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.

We consider systems of stochastic evolutionary equations of the $p$-Laplace type. We establish convergence rates for a finite-element based space-time approximation, where the error is measured in a suitable quasi-norm. Under natural regularity assumptions on the solution, our main result provides linear convergence in space and convergence of order $\alpha$ in time for all $\alpha\in(0,\frac{1}{2})$. The key ingredient of our analysis is a random time-grid, which allows us to compensate for the lack of time regularity. Our theoretical results are confirmed by numerical experiments.

To circumvent the ill-posedness issues present in various models of continuum fluid mechanics, we present a dynamical systems approach aiming at selection of physically relevant solutions. Even under the presence of infinitely many solutions to the full Euler system describing the motion of a compressible inviscid fluid, our approach permits to select a system of solutions (one trajectory for every initial condition) satisfying the classical semiflow property. Moreover, the selection respects the well accepted admissibility criteria for physical solutions, namely, maximization of the entropy production rate and the weak--strong uniqueness principle. Consequently, strong solutions are always selected whenever they exist and stationary states are stable and included in the selection as well. To this end, we introduce a notion of dissipative solution, which is given by a triple of density, momentum and total entropy defined as expectations of a suitable measure--valued solution.

An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the $p$-Laplace equation and system, with right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as H\"older, $\mathrm{BMO}$ and $\mathrm{VMO}$ spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $p=2$, and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in H\"older spaces, and complements the Jerison-Kenig gradient theory in Lebesgue spaces with a parallel in the oscillation spaces realm. The sharpness of our results is demonstrated by apropos examples.

It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill--posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to well--posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables - the mass density, the linear momentum, and the energy - in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.

We analyze the Markov property of solutions to the compressible Navier--Stokes system perturbed by a general multiplicative stochastic forcing. We show the existence of an almost sure Markov selection to the associated martingale problem. Our proof is based on the abstract framework introduced in [F. Flandoli, M. Romito: Markov selections for the 3D stochastic Navier--Stokes equations. Probab. Theory Relat. Fields 140, 407--458. (2008)]. A major difficulty arises from the fact, different from the incompressible case, that the velocity field is not continuous in time. In addition, it cannot be recovered from the variables whose time evolution is described by the Navier--Stokes system, namely, the density and the momentum. We overcome this issue by introducing an auxiliary variable into the Markov selection procedure.

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 pages

We study generalised Navier--Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent $p=p(\omega,t,x)$ (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies $p\geq p^->\frac{3n}{n+2}$ ($p^->1$ in two dimensions). Under additional assumptions we obtain also pathwise solutions.

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 pages

We study the full Navier--Stokes--Fourier system governing the motion of a general viscous, heat-conducting, and compressible fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise, (iii) random heat source in the internal energy balance. We establish existence of a weak martingale solution under physically grounded structural assumptions. As a byproduct of our theory we can show that stationary martingale solutions only exist in certain trivial cases.

In this paper, we consider the space $\mathrm{BV}^{\mathbb A}(\Omega)$ of functions of bounded $\mathbb A$-variation. For a given first order linear homogeneous differential operator with constant coefficients $\mathbb A$, this is the space of $L^1$--functions $u:\Omega\rightarrow\mathbb R^N$ such that the distributional differential expression $\mathbb A u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\Omega\subset\mathbb R^{n}$, $\mathrm{BV}^{\mathbb A}(\Omega)$-functions have an $L^1(\partial\Omega)$-trace if and only if $\mathbb A$ is $\mathbb C$-elliptic (or, equivalently, if the kernel of $\mathbb A$ is finite dimensional). The existence of an $L^1(\partial\Omega)$-trace was previously only known for the special cases that $\mathbb A u$ coincides either with the full or the symmetric gradient of the function $u$ (and hence covered the special cases $\mathrm{BV}$ or $\mathrm{BD}$). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the $\mathrm{BV}$- and $\mathrm{BD}$-setting) but rather compare projections onto the nullspace as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on $\mathbb A u$.

We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of weak (distributional) solutions. Specifically, we find a sequence $\tau_M \to \infty$ of positive stopping times for which the Euler system admits infinitely many solutions originating from the same initial data. The solutions are weak in the PDE sense but strong in the probabilistic sense, meaning, they are defined on an {\it a priori} given stochastic basis and adapted to the driving stochastic process.

We study the Navier--Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter's elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies $\gamma>\frac{12}{7}$ ($\gamma>1$ in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in [D. Lengeler, M. \Ruzicka, Weak Solutions for an Incompressible Newtonian Fluid Interacting with a Koiter Type Shell. Arch. Ration. Mech. Anal. 211 (2014), no. 1, 205--255] on incompressible Navier--Stokes equations.

We study the long-time behavior of solutions to a stochastically driven Navier-Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue--Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory. This can be seen as a kind of regularizing effect of the noise on the global-in-time solutions.