Related papers: Maximal regularity of backward difference time dis…
It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by a linear multistep method or Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize if the method…
In this paper we consider nonlinear parabolic systems with elliptic part which can be also degenerate. We prove optimal error estimates for smooth enough solutions. The main novelty, with respect to previous results, is that we obtain the…
In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first…
In this paper we present an abstract maximal $L^p$-regularity result up to $T = \infty$, that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional,…
In this note, we give an introduction to the concept of maximal $L^p$-regularity as a method to solve nonlinear partial differential equations. We first define maximal regularity for autonomous and non-autonomous problems and describe the…
Convergence results are shown for full discretizations of quasilinear parabolic partial differential equations on evolving surfaces. As a semidiscretization in space the evolving surface finite element method is considered, using a…
Assuming $A$ has maximal $L^p$-regularity, this paper investigates perturbations of $A$ by time-dependent operators $B$ that are unbounded and satisfy a critical $L^q$-integrability condition in time. We establish two main results. The…
We show convergence in the natural $L^{\infty}$- and $W^{1,\infty}$-norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this we show error estimates…
In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal $\ell^p$-regularity in…
We prove optimal regularity results in $L_p$-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is…
We prove maximal $L^p$-regularity for the stochastic evolution equation \[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial…
We consider a linear-quadratic pde constrained optimal control problem on an evolving surface with pointwise state constraints. We reformulate the optimization problem on a fixed surface and approximate the reformulated problem by a…
This article extends the semidiscrete maximal $L^p$-regularity results in [27] to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta}$, where $d$ is the…
In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the…
Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in…
A linear evolving surface partial differential equation is first discretized in space by an arbitrary Lagrangian Eulerian (ALE) evolving surface finite element method, and then in time either by a Runge-Kutta method, or by a backward…
The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent…
Space and time discretizations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usual L^2(\Omega)…
In this paper we prove maximal $L^p$-regularity for a system of parabolic PDEs, where the elliptic operator $A$ has coefficients which depend on time in a measurable way and are continuous in the space variable. The proof is based on…
We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this…