Maximal Regularity Result Research Articles

Maximal regularity for semilinear non-autonomous evolution equations in temporally weighted spaces

We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations u′(t)+A(t)u(t)=F(t,u),t-a.e.,u(0)=u0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\\, t \ ext {-a.e.}, \\, u(0)=u_0. \\end{aligned}$$\\end{document}Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space mathcal {H}. We prove the maximal regularity result in temporally weighted L^2-spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value u_0 and the inhomogeneous term F. Our results are motivated by boundary value problems.

An Lq(Lp)-theory for diffusion equations with space-time nonlocal operators

We present an Lq(Lp)-theory for the equation∂tαu=ϕ(Δ)u+f,t>0,x∈Rd;u(0,⋅)=u0. Here p,q>1, α∈(0,1), ∂tα is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: ∃δ0∈(0,1] and c>0 such that(0.1)c(Rr)δ0≤ϕ(R)ϕ(r),0<r<R<∞. We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove‖|∂tαu|+|u|+|ϕ(Δ)u|‖Lq([0,T];Lp)≤N(‖f‖Lq([0,T];Lp)+‖u0‖Bp,qϕ,2−2/αq), where Bp,qϕ,2−2/αq is a modified Besov space on Rd related to ϕ.Our approach is based on BMO estimate for p=q and vector-valued Calderón-Zygmund theorem for p≠q. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.

Maximal regularity result for a singular differential equation in the space of summable functions

Abstract We give sufficient conditions for the unique solvability and maximal regularity of a generalized solution of a second-order differential equation with unbounded diffusion, drift, and potential coefficients. We prove the compactness of the resolvent of the equation and an upper bound for the Kolmogorov widths of the set of solutions. It is assumed that the intermediate coefficient grows quickly and does not depend on the growth of potential. The diffusion coefficient is positive and can grow or disappear near infinity, i.e. the equation under consideration can degenerate. The study of such equation is motivated by applications in stochastic processes and financial mathematics.

Maximal regularity for non-autonomous Cauchy problems in weighted spaces

We consider the regularity for the non-autonomous Cauchy problem $$ u'(t) + A(t) u(t) = f(t)\quad (t \in [0, \tau]), \quad u(0) = u_0. $$ The time dependent operator A(t) is associated with (time dependent) sesquilinear forms on a Hilbert space \(\mathcal{H}\). We prove the maximal regularity result in temporally weighted L^2-spaces and other regularity properties for the solution of the problem under minimal regularity assumptions on the forms and the initial value u_0. Our results are motivated by boundary value problems.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/124/abstr.html

On a linearized Mullins-Sekerka/Stokes system for two-phase flows

&lt;p style='text-indent:20px;'&gt;We study a linearized Mullins-Sekerka/Stokes system in a bounded domain with various boundary conditions. This system plays an important role to prove the convergence of a Stokes/Cahn-Hilliard system to its sharp interface limit, which is a Stokes/Mullins-Sekerka system, and to prove solvability of the latter system locally in time. We prove solvability of the linearized system in suitable &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ L^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Sobolev spaces with the aid of a maximal regularity result for non-autonomous abstract linear evolution equations.&lt;/p&gt;

The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H∞-calculus on weighted Lp-spaces for power weights which fall outside the classical class of Ap-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

On Maximal Regularity for Abstract Parabolic Problems with Fractional Time Derivative

We consider initial value problems for abstract evolution equations with fractional time derivative. Concerning the Caputo derivative \(\mathbb {D}^\alpha u\), we show that certain assumptions, which are known to be sufficient to get a unique solution with a prescribed regularity, are also necessary. So we establish a maximal regularity result. We consider similar problems with the Riemann–Liouville derivative \(\partial ^\alpha u\). Here, we give a complete proof (necessity and sufficiency of the assumptions) of the corresponding maximal regularity results.

Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale $$ H ^{s-1}_D(\varOmega )$$ for $$s>0$$ sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces $$ H ^{s}(\varOmega )$$ . Ellipticity is enforced by assuming a Garding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.

Regularity of stochastic integral equations driven by Poisson random measures

We examine a stochastic integral equation driven by Poisson random measures. The increase or decrease of the regularity of the solution in space and time is examined as a function of the parameters of the kernels. The space regularity is measured in real interpolation spaces. The results generalize maximal regularity results obtained by Brzeźniak and Hausenblas.

Global existence and asymptotic behavior for a time fractional reaction–diffusion system

This paper is concerned with the existence of global in time solutions of a time fractional reaction–diffusion system with time fractional derivatives. Furthermore, the large time behavior of bounded solutions is investigated. Our method of proof relies on a maximal regularity result for fractional linear reaction–diffusion equations that has been derived by Bajlekova (2001).

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada–Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

ABSTRACTIn this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothe's method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.

A Priori Error Estimates for Three Dimensional Parabolic Optimal Control Problems with Pointwise Control

In this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise (Dirac-type) control in space on three-dimensional domains. The two-dimensional case was treated in [D. Leykekhman and B. Vexler, SIAM J. Numer. Anal., 51 (2013), pp. 2797--2821]; however, the three-dimensional case is technically much more involved. To approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish ${\mathcal O}(\sqrt{k}+h)$ order of convergence rate for the control in the $L^2$ norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, SIAM J. Control Optim., 52 (2014), pp. 97--119] and in addition does not require any relationship between the time step $k$ and the mesh size $h$. The main technical tools are discrete maximal parabolic regularity results and the best approximation-type estimate for the finite element error in the $L^\infty(\Omega;L^2(I))$ norm.

On some transmission problems set in a biological cell, analysis and resolution

Some transmission problems are set in bodies with a crown of small thickness ε>0. For instance, those concerning the conductivity in the biological cell. By a natural change of variables, we transform them in transmission problems set in two cylindrical bodies ]−∞,0[×]−π,π[ and ]0,δ[×]−π,π[ (where δ=ln⁡(1+ε)) and then, in some general elliptic abstract differential equations (Pδ)δ>0. The goal of this work is to give a complete study of these problems (Pδ)δ>0 for every δ>0. Existence, uniqueness and maximal regularity results are obtained for the classical solutions essentially by using the semigroups theory.

Weak Neumann implies $H^\infty$ for Stokes

Let $\Omega\subset {\mathbb R}^n$ be a domain with uniform $C^3$ boundary and assume that the Helmholtz decomposition exists in ${\mathbb L}^q(\Omega):=L^q(\Omega)^n$ for some $q\in(1,\infty)$. We show that a suitable translate of the Stokes operator admits a bounded ${\cal H}^\infty$-calculus in ${\mathbb L}_\sigma^p(\Omega)$ for $p\in(\min\{q,q'\},\max\{q,q'\})$. For the proof we use a recent maximal regularity result for the Stokes operator on such domains ([GHHS12]) and an abstract result for the ${\cal H}^\infty$-calculus in complemented subspaces ([KKW06], [KW13]).

Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension

We consider an elliptic Kolmogorov equationu Ku = f in a separable Hilbert space H. The Kolmogorov operator K is associated to an infinite dimensional convex gradient system: dX = (AX DU(X))dt + dW(t), where A is a self-adjoint operator in H and U is a convex lower semicontinuous function. Under mild assumptions we prove that for � > 0 and f 2 L 2 (H,�) the weak solution u belongs to the Sobolev space W 2,2 (H,�), whereis the log-concave probability measure of the system. Moreover maximal estimates on the gradient of u are proved. The maximal regularity results are used in the study of perturbed non gradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction-diffusion and Cahn-Hilliard stochastic PDEs.

Global existence and maximal regularity of solutions of gradient systems

In this article, we use a Galerkin method to prove a maximal regularity result for the following abstract gradient system { u ′ ( t ) + ∇ g ( t ) E ( u ( t ) ) = f ( t ) for a.e. t ∈ ( 0 , T ) , u ( 0 ) = u 0 . This abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients.

Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems

We prove local existence, uniqueness, Holder regularity in space and time, and smooth dependence in Holder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities in the equations as well as in the boundary conditions are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results (Griepentrog in Adv Differ Equ 12:781–840, 1031–1078, 2007) in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.

Non-homogeneous Navier-Stokes systems with order-parameter-dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order-parameter c. Moreover, an extra stress depending on c and ∇c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two-phase flow of viscous incompressible fluids. The so-called density-dependent Navier–Stokes system is also a special case of our system. We prove short-time existence of strong solution in Lq-Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $$({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], U(0) = u_0.\end{array}\right.$$ Here, \({(A(t))_{t\in [0,T]}}\) are unbounded operators with domains \({(D(A(t)))_{t\in [0,T]}}\) which may be time dependent. We assume that \({(A(t))_{t\in [0,T]}}\) satisfies the conditions of Acquistapace and Terreni. The functions F and B are nonlinear functions defined on certain interpolation spaces and \({u_0\in E}\) is the initial value. W H is a cylindrical Brownian motion on a separable Hilbert space H. We assume that the Banach space E is a UMD space with type 2. Under locally Lipschitz conditions we show that there exists a unique local mild solution of (SE). If the coefficients also satisfy a linear growth condition, then it is shown that the solution exists globally. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapien, and Zabczyk, to obtain space-time regularity results for the solution U of (SE). For Hilbert spaces E we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second-order stochastic partial differential equation which has been studied by Sanz-Sole and Vuillermot. This leads to several improvements of their result.