arXiv subjects

Dominic Breit

Publications and source records attributed to Dominic Breit.

At least 55 records · Page 3Linked to original sources

arXiv2019

Convergence rates for the numerical approximation of the 2D stochastic Navier-Stokes equations

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.

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arXiv2019

Generalized solutions to models of inviscid fluids

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.

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arXiv2019

Solution semiflow to the isentropic Euler system

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.

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arXiv2019

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 $(α_t, α_x)$-Hölder continuous.

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arXiv2019

Dissipative solutions and semiflow selection for the complete Euler system

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.

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arXiv2019

Global Schauder estimates for the $p$-Laplace system

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 Hölder, $\mathrm{BMO}$ 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 Hölder 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.

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arXiv2019

On the Trace Operator for Functions of Bounded $\mathbb{A}$-Variation

In this paper, we consider the space $\mathrm{BV}^{\mathbb A}(Ω)$ 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 $u:Ω\rightarrow\mathbb R^N$ such that the distributional differential expression $\mathbb A u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $Ω\subset\mathbb R^{n}$, $\mathrm{BV}^{\mathbb A}(Ω)$-functions have an $L^1(\partialΩ)$-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 $L^1(\partialΩ)$-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 $\mathbb A u$.

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arXiv2019

Electro-rheological fluids under random influences: martingale and strong solutions

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 $p=p(ω,t,x)$ (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.

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arXiv2019

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.

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arXiv2018

Markov selection for the stochastic compressible Navier--Stokes system

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.

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arXiv2017

Stochastic Navier-Stokes-Fourier equations

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.

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arXiv2017

On solvability and ill-posedness of the compressible Euler system subject to stochastic forces

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 $τ_M \to \infty$ 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.

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arXiv2017

Compressible fluids interacting with a linear-elastic shell

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 $γ>\frac{12}{7}$ ($γ>1$ 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.

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arXiv2017

Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces

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.

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arXiv2017

Trace-free Korn inequalities in Orlicz spaces

Necessary and sufficient conditions are exhibited for a Korn type inequality to hold between (possibly different) Orlicz norms of the gradient of vector-valued functions and of the deviatoric part of their symmetric gradients. As a byproduct of our approach, a positive answer is given to the question of the necessity of the same sufficient conditions in related Korn type inequalities for the full symmetric gradient, for negative Orlicz-Sobolev norms, and for the gradient of the Bogovskii operator.

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arXiv2016

Regularity theory for nonlinear systems of SPDEs

We consider systems of stochastic evolutionary equations of the type $$du=\mathrm{div}\,S(\nabla u)\,dt+Φ(u)dW_t$$ where $S$ is a non-linear operator, for instance the $p$-Laplacian $$S(ξ)=(1+|ξ|)^{p-2}ξ,\quad ξ\in\mathbb R^{d\times D},$$ with $p\in(1,\infty)$ and $Φ$ grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity: $$\mathbb E\bigg[\sup_{t\in(0,T)}\int_{G'}|\nabla u(t)|^2\,dx+\int_0^T\int_{G'}|\nabla F(\nabla u)|^2\,dx\,dt\bigg]<\infty,$$ where $F(ξ)=(1+|ξ|)^{\frac{p-2}{2}}ξ$. If we have Uhlenbeck-structure then $\mathbb E\big[\|\nabla u\|_q^q\big]$ is finite for all $q<\infty$.

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arXiv2016

A variational approach to solitary gravity-capillary interfacial waves with infinite depth

We present an existence and stability theory for gravity-capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy $\mathcal{E}$ subject to the constraint $\mathcal{I}=2μ$, where $\mathcal{I}$ is the wave momentum and $0< μ< μ_0$, where $μ_0$ is chosen small enough for the validity of our calculations. Since $\mathcal{E}$ and $\mathcal{I}$ are both conserved quantities a standard argument asserts the stability of the set $D_μ$ of minimisers: solutions starting near $D_μ$ remain close to $D_μ$ in a suitably defined energy space over their interval of existence. The solitary waves which we construct are of small amplitude and are to leading order described by the cubic nonlinear Schrödinger equation. They exist in a parameter region in which the `slow' branch of the dispersion relation has a strict non-degenerate global minimum and the corresponding nonlinear Schrödinger equation is of focussing type. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the model equation as $μ\downarrow 0$.

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arXiv2016

Weak Solutions for a Non-Newtonian Diffuse Interface Model with Different Densities

We consider weak solutions for a diffuse interface model of two non-Newtonian viscous, incompressible fluids of power-law type in the case of different densities in a bounded, sufficiently smooth domain. This leads to a coupled system of a nonhomogenouos generalized Navier-Stokes system and a Cahn-Hilliard equation. For the Cahn-Hilliard part a smooth free energy density and a constant, positive mobility is assumed. Using the $L^\infty$-truncation method we prove existence of weak solutions for a power-law exponent $p>\frac{2d+2}{d+2}$, $d=2,3$.

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