Maximal regularity for non-autonomous stochastic evolution equations in UMD Banach spaces

A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partial differential equation.     

Maximal regularity for non-autonomous Schrödinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations t_∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m.     

Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation $\text{u′(t) ? A(t) u(t) = ?(t), t ∈ [0, T]}$, with the initial datum u(0) = x, in the space C([0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces D_A(0)(θ, ∞) and D_A(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given.     

Maximal time regularity for degenerate evolution integro-differential equations

We provide maximal time regularity properties for the solutions to a class of degenerate first-order integro-differential Cauchy problems in a Banach space X. In particular, we show that an additional condition of space regularity on the data it suffices for restoring the loss of time regularity which arises naturally when dealing with the degenerate case.     

Stability for non-autonomous linear evolution equations with L p -maximal regularity

We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem $$(P),\left\{ \begin{gathered} \dot u(t) + A(t)u(t) = f(t) t - a.e. on [0,\tau ] \hfill \\ u(0) = 0, \hfill \\ \end{gathered} \right.$$ where A: [0, τ] → L(X,D) is a bounded and strongly measurable function and X, D are Banach spaces such that . Our main concern is to characterize L ^p -maximal regularity and to give an explicit approximation of the problem (P).     

Maximal regularity of evolution equations on discrete time scales

We consider the question of Lp-maximal regularity for inhomogeneous Cauchy problems in Banach spaces using operator-valued Fourier multipliers. This follows results by L. Weis in the continuous time setting and by S. Blunck for discrete time evolution equations. We generalize the later result to the case of some discrete time scales (discrete problems with nonconstant step size). First we introduce an adequate evolution family of operators to consider the general problem. Then we consider the case where the step size is a periodic sequence by rewriting the problem on a product space and using operator matrix valued Fourier multipliers. Finally we give a perturbation result allowing to consider a wider class of step sizes.     

Operator splitting for non-autonomous evolution equations

We establish general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tools are evolution semigroups allowing the direct application of existing results on autonomous problems. The results obtained are illustrated by the example of an autonomous diffusion equation perturbed with time dependent potential. We also prove convergence rates for the sequential splitting applied to this problem.     

Hölder regularity for non-autonomous abstract parabolic equations

This paper contains some existence and uniqueness results for the strict and classical solutionsu : [0,T] →E of the non-autonomous evolution equationu ¹(t)=Λ(t)u(t)+f(t) in a Banach spaceE under the classical Tanabe-Sobolevski assumptions. These results do not require use of the fundamental solution and give new information about the hölder-regularity of the solutions.     

Maximal L p regularity for parabolic difference equations

In this paper, we consider a family of finite difference operators {Ah }_{h >0} on discrete L _q ‐spaces L _q (?^N _h ). We show that the solution u _h to u ′_h (t) – A _h u _h(t) = f _h (t), t > 0, u _h (0) = 0 satisfies the estimate ‖A _h u _h ‖equation/tex2gif-inf-15.gif ≤ C ‖f _h ‖equation/tex2gif-inf-21.gif, where C is independent of h and f _h . In this case, the family {A _h }_{h >0} is said to have discrete maximal L _p regularity on the discrete L _q ‐space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some existence and regularity results for abstract non-autonomous parabolic equations

We study existence, uniqueness and regularity of the strict, classical and strong solutions $\text{u? C(¦0, T ¦,E)}$ of the non-autonomous evolution equation $\text{u′(t) ? A(t)u(t)=?(t)}$, with the initial datum u(0) = x, in a Banach space E, under the classical Kato-Tanabe assumptions. The domains of the operators A(t) are not needed to be dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solution and its derivative.     

Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations

In this paper, we show that a pseudo-differential operator associated to a symbol ( being a Hilbert space) which admits a holomorphic extension to a suitable sector of acts as a bounded operator on . By showing that maximal -regularity for the non-autonomous parabolic equation is independent of , we obtain as a consequence a maximal -regularity result for solutions of the above equation.

     

Existence and sharp regularity results for linear parabolic non-autonomous integro-differential equations

We study existence, uniqueness and maximal regularity of the strict solutionu∈C ¹([0,T],E) of the integro-differential equation \(u'(t) - A(t)u(t) - \int {_0^1 } B(t,s)u(s)ds = f(t),t \in [0,T],\) with the initial datumu(0)=x, in a Banach spaceE, {itA(itt)}_f∈|0,1| is a family of generators of analytic semigroups whose domainsD _A(t) are not constant int as well as (possibly) not dense inE, whereas {itB(itt)}_0≦11≦T is a family of closed linear operators withD _B(t,s) ?D _A(s) ∨t∈[s, T]. We prove necessary and sufficient conditions for existence of the strict solution and for Hölder continuity of its derivative; well-posedness of the problem with respect to the Hölder norms is also shown.     

Maximal regularity of second-order equations with delay

We characterize existence and uniqueness of solutions of an inhomogeneous abstract delay equation in Hölder spaces. The method is based on the theory of operator-valued Fourier multipliers.     

Maximal regularity for perturbed integral equations on periodic Lebesgue spaces

We characterize the maximal regularity of periodic solutions for an additive perturbed integral equation with infinite delay in the vector-valued Lebesgue spaces. Our method is based on operator-valued Fourier multipliers. We also study resonances, characterizing the existence of solutions in terms of a compatibility condition on the forcing term.     

Maximal regularity of type L p for abstract parabolic Volterra equations

We prove maximal regularity results of type L_p for abstract parabolic Volterra equations including problems with inhomogeneous boundary data. Our approach is purely operator theoretic. It uses the inversion of the convolution, the Dore-Venni theorem, the Mikhlin theorem in the operator-valued version, and real interpolation. Known results on L_p-regularity of abstract Cauchy problems and abstract parabolic pdes with inhomogeneous boundary conditions are recovered. As an application we consider the heat equation of memory type with inhomogeneous boundary condition.     

Extrapolation spaces and minimal regularity for evolution equations

A new approach to extrapolation spaces for unbounded linear operators is applied to evolution equations in a Banach space in order to derive existence and properties of its solutions under minimal assumptions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday