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Dominic Breit

Publications and source records attributed to Dominic Breit.

64 records · Page 4Linked to original sources
arXiv2016

Local strong solutions to the stochastic compressible Navier-Stokes system

We study the Navier-Stokes system describing the motion of a compressible viscous fluid driven by a nonlinear multiplicative stochastic force. We establish local in time existence (up to a positive stopping time) of a unique solution, which is strong in both PDE and probabilistic sense. Our approach relies on rewriting the problem as a symmetric hyperbolic system augmented by partial diffusion, which is solved via a suitable approximation procedure using the stochastic compactness method and the Yamada-Watanabe type argument based on the Gyöngy-Krylov characterization of convergence in probability. This leads to the existence of a strong (in the PDE sense) pathwise solution. Finally, we use various stopping time arguments to establish the local existence of a unique strong solution to the original problem.

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arXiv2016

The A-Stokes approximation for non-stationary problems

Let $\mathcal A$ be an elliptic tensor. A function $v\in L^1(I;LD_{div}(B))$ is a solution to the non-stationary $\mathcal A $-Stokes problem iff \begin{align}\label{abs} \int_Q v\cdot\partial_tϕ\,dx\,dt-\int_Q \mathcal A(\varepsilon(v),\varepsilon(ϕ))\,dx\,dt=0\quad\forallϕ\in C^{\infty}_{0,div}(Q), \end{align} where $Q:=I\times B$, $B\subset\mathbb R^d$ bounded. If the l.h.s. is not zero but small we talk about almost solutions. We present an approximation result in the fashion of the $\mathcal A$-caloric approximation for the non-stationary $\mathcal A $-Stokes problem. Precisely, we show that every almost solution $v\in L^p(I;W^{1,p}_{div}(B))$, $1<p<\infty$, can be approximated by a solution in the $L^s(I;W^{1,s}(B))$-sense for all $s<p$. So, we extend the stationary $\mathcal A$-Stokes approximation by Breit-Diening-Fuchs to parabolic problems.

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arXiv2016

Incompressible limit for compressible fluids with stochastic forcing

We study the asymptotic behavior of the isentropic Navier-Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier-Stokes system is identified as the limit problem.

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arXiv2015

Stochastic Navier-Stokes equations for compressible fluids

We study the Navier-Stokes equations governing the motion of isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density, which is affine linear in momentum and satisfies suitable growth assumptions with respect to density, and establish existence of the so-called finite energy weak martingale solution under the condition that the adiabatic constant satisfies $γ>3/2$. The proof is based on a four layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.

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arXiv2015

Compressible fluids driven by stochastic forcing: The relative energy inequality and applications

We show the relative energy inequality for the compressible Navier-Stokes system driven by a stochastic forcing. As a corollary, we prove the weak-strong uniqueness property (pathwise and in law) and convergence of weak solutions in the inviscid-incompressible limit. In particular, we establish a Yamada-Watanabe type result in the context of the compressible Navier-Stokes system, that is, pathwise weak--strong uniqueness implies weak--strong uniqueness in law.

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arXiv2015

Pointwise Calderón-Zygmund gradient estimates for the $p$-Laplace system

Pointwise estimates for the gradient of solutions to the $p$-Laplace system with right-hand side in divergence form are established. They enable us to develop a nonlinear counterpart of the classical Calderón-Zygmund theory in terms of Calderón-Zygmund singular integrals, for the Laplacian. As a consequence, a flexible, comprehensive approach to gradient bounds for the $p$-Laplace system for a broad class of norms is derived. In particular, new gradient estimates are exhibited, and well-known results in customary function spaces are easily recovered.

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arXiv2015

On time regularity of stochastic evolution equations with monotone coefficients

We report on a time regularity result for stochastic evolutionary PDEs with monotone coefficients. If the diffusion coefficient is bounded in time without additional space regularity we obtain a fractional Sobolev type time regularity of order up to $\tfrac{1}{2}$ for a certain functional $ G( u )$ of the solution. Namely, $ G( u )=\nabla u $ in the case of the heat equation and $G( u )=|\nabla u |^{\frac{p-2}{2}}\nabla u $ for the $p$-Laplacian. The motivation is twofold. On the one hand, it turns out that this is the natural time regularity result that allows to establish the optimal rates of convergence for numerical schemes based on a time discretization. On the other hand, in the linear case, i.e. where the solution is given by a stochastic convolution, our result complements the known stochastic maximal space-time regularity results for the borderline case not covered by other methods.

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arXiv2015

Existence theory for stochastic power law fluids

We consider the equations of motion for an incompressible Non-Newtonian fluid in a bounded Lipschitz domain $G\subset\mathbb R^d$ during the time intervall $(0,T)$ together with a stochastic perturbation driven by a Brownian motion $W$. The balance of momentum reads as $$dv=\mathrm{div}\, S\,dt-(\nabla v)v\,dt+\nablaπ\,dt+f\,dt+Φ(v)\,dW_t,$$ where $v$ is the velocity, $π$ the pressure and $f$ an external volume force. We assume the common power law model $S(\varepsilon(v))=\big(1+|\varepsilon(v)|\big)^{p-2} \varepsilon(v)$ and show the existence of weak (martingale) solutions provided $p>\tfrac{2d+2}{d+2}$. Our approach is based on the $L^\infty$-truncation and a harmonic pressure decomposition which are adapted to the stochastic setting.

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arXiv2014

Negative Orlicz-Sobolev norms and strongly nonlinear systems in fluid mechanics

We prove a version of the negative norm theorem in Orlicz-Sobolev spaces. A study of continuity properties of the Bogovskii -operator between Orlicz spaces is a crucial step, of independent interest, in our approach. Applications to the problem of pressure reconstruction for Non-Newtonian fluids governed by constitutive laws, which are not necessarily power type, are presented. A key inequality for a numerical analysis of the underlying elliptic system is also derived.

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