On the interior regularity of weak solutions to the non-stationary Stokes system

In this paper, we prove that any weak solution to the non-stationary Stokes system in 3D with right hand side -div f satisfying (1.4) below, belongs to C( ]0, T[; C ^α (Ω)). The proof is based on Campanato-type inequalities and the existence of a local pressure introduced in Wolf [13]. Proc. Conference “Variational analysis and PDE’s”. Intern. Centre “E. Majorana”, Erice, July 5–14, 2006.     

Time stepping procedures for the non-stationary Stokes equations

We consider first and second-order implicit time stepping procedures for the non-stationary Stokes equations in bounded domains of ?³. Using energy estimates we prove the optimal convergence properties in the Sobolev spaces H^m(G)(m = 0, 1, 2) uniformly in time, provided that the Stokes solution has a certain degree of regularity. Here in the case of the second-order scheme (method of Crank–Nicholson) the Stokes solution has to satisfy a non-local compatibility condition at the initial time t = O, which can be satisfied by a special initial construction.     

Uniform regularity and vanishing viscosity limit for the chemotaxis‐Navier‐Stokes system in a 3D bounded domain

We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis‐Navier‐Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis‐Navier‐Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis‐Euler system.     

Very weak solutions for the Stokes problem in an exterior domain

In this paper, we study the Stokes problem in exterior domain of ${\mathbb{R}^{3}}$ . We are interested in the existence and the uniqueness of very weak solutions. Here, we extend a result proved by Farwig et al. (J Math Soc Japan 59(1):127–150, 2007) and we prove the existence and the uniqueness of a second type of very weak solution.     

Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Maximal regularity for a free boundary problem

This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcy's law.We prove existence of a unique maximal classical solution, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration.Supported by Schweizerischer Nationalfonds     

On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

A regularity result for the Stokes problem in a convex polygon

In this paper we show that the solution of the Stokes problem has square integrable second derivatives provided that the domain is a convex polygon in the plane and the nonhomogeneous term is square integrable.     

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.     

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)     

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.