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search.filters.applied.f.access: openAccess × Subject: maximal regularity ×

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    On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) ter Elst, A.F.M.; Haller-Dintelmann, Robert; Rehberg, Joachim; Tolksdorf, Patrick
    Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on Lp(Ω). Additional properties like analyticity of the semigroup, H∞-calculus and maximal regularity arealso discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of p's for small imaginary parts of the coefficients. Our results are based on the recent notion of p-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.
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    Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Druet, Pierre-Étienne
    We consider a Navier--Stokes--Fick--Onsager--Fourier system of PDEs describing mass, energy and momentum balance in a Newtonian fluid with composite molecular structure. For the resulting parabolic-hyperbolic system, we introduce the notion of optimal regularity of mixed type, and we prove the short-time existence of strong solutions for a typical initial boundary-value-problem. By means of a partial maximum principle, we moreover show that such a solution cannot degenerate in finite time due to blow-up or vanishing of the temperature or the partial mass densities. This second result is however only valid under certain growth conditions on the phenomenological coefficients. In order to obtain some illustration of the theory, we set up a special constitutive model for volume-additive mixtures.
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    Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Griepentrog, Jens A.
    This text is devoted to maximal regularity results for second order parabolic systems on Lipschitz domains of space dimension n ≥ 3 with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of Sobolev–Morrey spaces for solutions and right hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients.
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    Quasilinear parabolic systems with mixed boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Hieber, Matthias; Rehberg, Joachim
    In this paper we investigate quasilinear systems of reaction-diffusion equations with mixed Dirichlet-Neumann bondary conditions on non smooth domains. Using techniques from maximal regularity and heat-kernel estimates we prove existence of a unique solution to systems of this type.
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    Local existence, uniqueness, and smooth dependence for nonsmooth quasilinear parabolic problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Griepentrog, Jens André; Recke, Lutz
    A general theory on local existence, uniqueness, regularity, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data has been developed. 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 the abstract formulations of the initial boundary value problems, has been closed. The main tools are new maximal regularity results of the first author in Sobolev-Morrey spaces, linearization techniques and the Implicit Function Theorem. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, chemotactic aggregation in heterogeneous environments as well as optimal control by means of quasilinear elliptic and parabolic PDEs with nonsmooth data.