arXiv subjects

Sebastian Schwarzacher

Publications and source records attributed to Sebastian Schwarzacher.

50 records · Page 3Linked to original sources

arXiv2017

Regularity and approximation of strong solutions to rate-independent systems

Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this work we prove the existence of Hölder-regular strong solutions for a class of rate-independent systems. We also establish additional higher regularity results that guarantee the uniqueness of strong solutions. The proof proceeds via a time-discrete Rothe approximation and careful elliptic regularity estimates depending in a quantitative way on the (local) convexity of the potential featuring in the model. In the second part of the paper we show that our strong solutions may be approximated by a fully discrete numerical scheme based on a spatial finite element discretization, whose rate of convergence is consistent with the regularity of strong solutions whose existence and uniqueness are established.

math.AP↗

arXiv2017

Self-improving property of degenerate parabolic equations of porous medium-type

We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse Hölder inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calderón-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution $u$.

math.AP↗

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.

math.AP↗

arXiv2017

Homogenization of the compressible Navier-Stokes equations in domains with very tiny holes

We consider the homogenization problem of the compressible Navier-Stokes equations in a bounded three dimensional domain perforated with very tiny holes. As the number of holes increases to infinity, we show that, if the size of the holes is small enough, the homogenized equations are the same as the compressible Navier-Stokes equations in the homogeneous domain---domain without holes. This coincides with the previous studies for the Stokes equations and the stationary Navier-Stokes equations. It is the first result of this kind in the instationary barotropic compressible setting. The main technical novelty is the study of the Bogovskii operator in non-Lipschitz domains.

math.AP↗

arXiv2016

A unified theory for some non Newtonian fluids under singular forcing

We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain non-degenerate power-law or Carreau-type fluids. We provide the full-range theory, namely existence, optimal regularity and uniqueness of solutions, not only with respect to forcing belonging to Lebesgue spaces, but also with respect to their refinements, namely the weighted Lebesgue spaces, with weights in a respective Muckenhoupt class. The analytical highlight is derivation of existence and uniqueness theory for forcing with its regularity well-below the natural duality exponent, via estimates in weighted spaces. It is a generalization of [Bulíček, Diening, Schwarzacher] to incompressible fluids. Moreover, two technical results, needed for our analysis, may be useful for further studies. They are: the solenoidal, weighted, biting div-curl lemma and the solenoidal Lipschitz approximations on domains.

math.AP↗

arXiv2016

Existence and regularity for a class of rate-independent systems

Despite the many applications of rate-independent systems, their regularity theory is still largely unexplored. Usually, only weak solution with potentially very low regularity are considered, which requires non-smooth techniques. In this work, however, we directly prove the existence of Hölder-regular strong solutions for a class of rate-independent systems. We also establish further assertions about higher regularity of our solutions. The proof proceeds via a time-discrete Rothe approximation, careful elliptic regularity estimates in the discrete situation and evolutionary techniques.

math.AP↗

arXiv2016

Existence of very weak solutions to elliptic systems of p-Laplacian type

We study vector valued solutions to non-linear elliptic partial differential equations with $p$-growth. Existence of a solution is shown in case the right hand side is the divergence of a function which is only $q$ integrable, where $q$ is strictly below but close to the duality exponent $p'$. It implies that possibly degenerate operators of $p$-Laplacian type are well posed in a larger class then the natural space of existence. The key novelty here is a refined a priori estimate, that recovers a duality relation between the right hand side and the solution in terms of weighted Lebesgue spaces.

math.AP↗

arXiv2016

Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems

We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are sufficient and in many cases also necessary for building such a theory. We provide a unified approach that leads qualitatively to the same theory as that one available for linear elliptic problems with continuous coeffcients, e.g. the Poisson equation. The result is based on several novel tools that are of independent interest: local and global estimates for (non)linear elliptic systems in weighted Lebesgue spaces with Muckenhoupt weights, a generalization of the celebrated div{curl lemma for identification of a weak limit in border line spaces and the introduction of a Lipschitz approximation that is stable in weighted Sobolev spaces.

math.AP↗

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.

math.AP↗

arXiv2015

On regularity of the time derivative for degenerate parabolic systems

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau type. By the use of special weak different quotients, adapted to the variational structure we bound fractional derivatives of $u_t$ in time and space direction. Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic p-Laplace equation.

math.AP↗

arXiv2013

Global gradient estimates for the $p(\cdot)$-Laplacian

We consider Calderón-Zygmund type estimates for the non-homogeneous $p(\cdot)$-Laplacian system $ -\text{div}(|D u|^{p(\cdot)-2} Du) = -\text{div}(|G|^{p(\cdot)-2} G),$ where $p$ is a variable exponent. We show that $|G|^{p(\cdot)} \in L^q(\mathbb{R}^n)$ implies $|D u|^{p(\cdot)} \in L^q(\mathbb{R}^n)$ for any $q \geq 1$. We also prove local estimates independent of the size of the domain and introduce new techniques to variable analysis.

math.AP↗

arXiv2013

Hölder-Zygmund Estimates for Degenerate Parabolic Systems

We consider energy solutions of the inhomogeneous parabolic $p$-Laplacien system $\partial_t u-\text{div}(|D u|^{p-2}D u)=-\text{div} g$. We show in the case $p\geq 2$ that if the right hand side $g$ is locally in $L^\infty(\text{BMO})$, then $u$ is locally in $L^\infty(\mathcal{C}^1)$, where $\mathcal{C}^1$ is the 1-Hölder--Zygmund space. This is the borderline case of the Calderón-Zygmund theorey. We provide local quantitative estimates. We also show that finer properties of $g$ are conserved by $D u$, e.g.\ Hölder continuity. Moreover, we prove a new decay for gradients of $p$-caloric solutions for all $\frac{2n}{n+2}<p<\infty$.

math.AP↗

arXiv2013

Convex Hull Property and Maximum Principle for Finite Element Minimisers of General Convex Functionals

The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for $¶_1$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the $p$-Laplacian and the mean curvature problem. In the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.

math.NA↗

arXiv2012

Campanato estimates for the generalized Stokes System

We study interior regularity of solutions of a generalized stationary Stokes problem in the plane. The main, elliptic part of the problem is given in the form div(A(Du)), where D is the symmetric part of the gradient. The model case is A(Du)=(kappa+|Du|)^{p-2}Du. We show optimal BMO and Campanato estimates for A(Du). Some corollaries for the generalized stationary Navier-Stokes system and for its evolutionary variant are also mentioned.

math.AP↗