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Start date: 1999 × Type: Text × Type: report × Version: publishedVersion × Subject: regularity × Subject: uniqueness ×

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    Resolvent estimates in W-1,p related to strongly coupled linear parabolic systems with coupled nonsmooth capacities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Glitzky, Annegret; Hünlich, Rolf
    We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in W-1,p spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain Lp estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result.
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    Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Patterson, Robert I.A.
    Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of d-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, textitStoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.
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    Regularity and uniqueness in quasilinear parabolic systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Krejčí, Pavel; Panizzi, Lucia
    Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.
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    A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Sprekels, Jürgen; Valdinoci, Enrico
    In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the sth power of a positive definite operator having a discrete spectrum in R+. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter s. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter s serves as the control parameter that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new class of identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter s also the domain of definition, and thus the underlying function space, of the fractional operator changes.