arXiv subjects

D. Breit

Publications and source records attributed to D. Breit.

6 recordsLinked to original sources

arXiv2026

Well-posedness and regularity in thermodynamics of compressible fluid-structure interactions

We consider the interaction of a general viscous compressible and heat-conducting fluid with an elastic shell located at the boundary of the fluid's domain. The fluid is described by the compressible Navier-Stokes-Fourier equations and the shell evolves in accordance with a viscoelastic beam equation. Both are coupled through kinematic boundary conditions and the balance of forces. Our first result is the local-in-time well-posedness of the underlying coupled system of nonlinear PDEs in smooth function spaces. Eventually, we prove a conditional regularity criterion which is in the spirit of the Beale-Kato-Majda condition for the incompressible Euler equations combined with the boundedness of density and temperature. It rests upon a weak-strong uniqueness result for the underlying system.

math.AP↗

arXiv2026

A Higher Order Discretization for the Stochastic Navier--Stokes equations with additive Noise

We propose a new higher-order time discretization scheme for the stochastic Navier--Stokes equations with additive noise, where its velocity and pressure approximates converge at strong rate $1.5$ in probability. The construction rests on its reformulation as a random PDE for the transform $y = u- \Phi W$, and different higher order numerical quadrature rules for the diffusion and the drift part. The theoretical findings are supported by numerical simulations.

math.NA↗

arXiv2025

Riesz potential estimates under co-canceling constraints

Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms ``far" from $L^1$, but to be weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Rai\c{t}\u{a} and Spector, and by Stolyarov proved that this gap can be filled in Riesz potential inequalities for vector-valued functions in $L^1$ fulfilling a co-canceling differential condition. The present work demonstrates that such a property is not just peculiar to the space $L^1$. As a consequence, Riesz potential inequalities under the co-canceling constraint are offered for general families of rearrangement-invariant spaces, such as the Orlicz spaces and the Lorentz-Zygmund spaces. Especially relevant instances of inequalities for domain spaces neighboring $L^1$ are singled out.

math.FA↗

arXiv2025

Compressible fluids excited by space-dependent transport noise

We study the compressible Navier-Stokes system driven by physically relevant transport noise, where the noise influences both the continuity and momentum equations. Our approach is based on transforming the system into a partial differential equation with random, time- and space-dependent coefficients. A key challenge arises from the fact that these coefficients are non-differentiable in time, rendering standard compactness arguments for the identification of the pressure inapplicable. To overcome this difficulty, we develop a novel multi-layer approximation scheme and introduce a precise localization strategy with respect to both the sample space and time variable. The limit pressure is then identified via the corresponding effective viscous flux identity. By means of stochastic compactness methods, particularly Skorokhod's representation theorem and its generalization by Jakubowski, we ensure the progressive measurability required to return to the original system. Our results broaden the applicability of transport noise models in fluid dynamics and offer new insights into the interaction between stochastic effects and compressibility.

math.AP↗

arXiv2013

Finite element approximation of the $p(\cdot)$-Laplacian

We study a~priori estimates for the Dirichlet problem of the $p(\cdot)$-Laplacian, \[-\mathrm{div}(|\nabla v|^{p(\cdot)-2} \nabla v) = f. \] We show that the gradients of the finite element approximation with zero boundary data converges with rate $O(h^\alpha)$ if the exponent $p$ is $\alpha$-H\"{o}lder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e. we measure the $L^2$-error of $|\nabla v|^{\frac{p-2}{2}} \nabla v$.

math.NA↗

arXiv2013

Solenoidal Lipschitz truncation for parabolic PDE's

We consider functions $u\in L^\infty(L^2)\cap L^p(W^{1,p})$ with $1<p<\infty$ on a time space domain. Solutions to non-linear evolutionary PDE's typically belong to these spaces. Many applications require a Lipschitz approximation $u_λ$ of $u$ which coincides with $u$ on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids in [DRW10]. Since ${\rm div} u_λ=0$, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of [BDF12].

math.AP↗