We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal $L_p$–regularity for the generator of an analytic semigroup $T_t$ in terms of the R–boundedness of the resolvent of A or the semigroup $T_t$.

Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

In the modeling of collective effects arising in bacterial suspensions in fluid drops, coupled chemotaxis-(Navier–)Stokes systems generalizing the prototype have been proposed to describe the spont

Global Large-Data Solutions in a Chemotaxis-(Navier–)Stokes System Modeling Cellular Swimming in Fluid Drops

We develop a very general operator-valued functional calcu- lus for operators with an H 1 −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H 1 calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Ba- nach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting.

The H ∞ −calculus and sums of closed operators

In this paper we investigate vector-valued parabolic initial boundary value problems ${(\mathcal A(t,x,D)}$ , ${\mathcal B_j(t,x,D))}$ subject to general boundary conditions in domains G in ${\mathbb R^n}$ with compact C 2m -boundary. The top-order coefficients of ${\mathcal A}$ are assumed to be continuous. We characterize optimal L p -L q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on ${\mathcal A}$ and the Lopatinskii–Shapiro condition on ${(\mathcal A, \mathcal B_1,\dots, \mathcal B_m)}$ are necessary for these L p -L q -estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.

Optimal L p - L q -estimates for parabolic boundary value problems with inhomogeneous data

We prove the global existence, along with some basic boundedness properties, of weak solutions to a PDE-ODE system modeling the multiscale invasion of tumor cells through the surrounding tissue matrix. The model has been proposed in [G. Meral, C. Stinner, and C. Surulescu, On a Multiscale Model Involving Cell Contractivity and its Effects on Tumor Invasion, preprint, TU Kaiserslautern, Kaiserslautern, Germany, 2013] and accounts on the macroscopic level for the evolution of cell and tissue densities, along with the concentration of a chemoattractant, while on the subcellular level it involves the binding of integrins to soluble and insoluble components of the peritumoral region. The connection between the two scales is realized with the aid of a contractivity function characterizing the ability of the tumor cells to adapt their motility behavior to their subcellular dynamics. The resulting system, consisting of three partial and three ordinary differential equations including a temporal delay, in particul...

https://kluedo.ub.uni-kl.de/files/3621/ssw15.pdf

Global Weak Solutions in a PDE-ODE System Modeling Multiscale Cancer Cell Invasion

Let A be the generator of an analytic semigroup Ton L2(Ω), where Ω is a homogeneous space with doubling property. We prove maximal Lp-Lp a—priori estimates for the solution of the parabolic evolution equation u'(t)=Au(t)+f(t), u(0)=0 provided Tmay be represented by a heat—kernel satisfying certain bounds (and in particular a Gaussian bound). 1991 Mathematics Subject Classification:35K22, 58D25, 47D06

Hieber Matthias

Papers

Heat kernels and maximal lp—lqestimates for parabolic evolution equations