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Marcus Waurick

Publications and source records attributed to Marcus Waurick.

At least 19 recordsLinked to original sources
arXiv2026

Revisiting Picard's proof of the Picard--Weber--Weck selection theorem

Revisiting the rationale provided by Picard in the seminal paper [13], we provide an independent, self-contained proof of the Maxwell compactness property for weak Lipschitz domains in the Euclidean setting, detouring technical complications like the differential forms setting, Gaffney's inequality for smooth domains, Calderon's extension theorem or regularity theory for PDEs, or systems thereof, on smooth domains that are used in this context. As a by-product we provide an independent proof of the classical Gaffney estimate for cubes using only $L^2$-completeness of the Fourier bases.

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arXiv2026

Nonlinear Media via Nonlocal Homogenisation

We consider a nonlinear PDE describing a nonlinear electrostatic medium with nonlocal dielectricity. The existence proof for the corresponding equation is based on Schauder's theorem and a new compactness theorem for moving coefficients (``Helga's Theorem''). This technique uses insights from (operator-theoretic/topological) homogenisation theory. Surprisingly, even though monotonicity assumptions are neither used nor valid, the underlying domain is only required to be weak Lipschitz and no assumption on the derivatives of the nonlinearity is needed.

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arXiv2026

The Free Lunch Theorem of Homogenisation

We show that H-convergence for multiplication type operators as envisioned by Murat and Tartar in the 1970's always implies nonlocal H-convergence as introduced in 2018 in Calc.~Var.~PDE 57(6):159. In contrast to earlier findings, the results presented here work for arbitrary space dimensions, are not bound to a certain geometry of the underlying domain, and do not explicitly require an underlying Hilbert complex for the application of any particular version of the div-curl lemma. We extend classical theory and the main results to more general differential operators with different boundary conditions and orders. Furthermore, the present results confirm homogenisation formulas used in the literature of which we failed to find an explicit proof. As a consequence, H-convergence for multiplication operators in divergence form problems will always imply H-type convergence for a different variational problem for free.

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arXiv2026

Nonautonomous systems of evolution inclusions

We prove the existence of global solutions for some coupled systems of partially nonautonomous evolution inclusions comprised of a Cauchy problem with a compact resolvent semigroup generator and an evolution equation governed by a subdifferential of a real potential. Our system in particular includes nonautonomous generalized Schr\"odinger-Debye systems of inclusions with variable exponents, but extends to hyperbolic-parabolic systems of inclusions in particular to Maxwell-parabolic systems of inclusions. Methodologically, we extend an approach of Vrabie et al. to the nonautonomous case and make use of standard semigroup tools to accomodate non-parabolic behaviour of solutions paired with a new existence result for measurable selections. The combination of the latter two requires the set-valued coupling terms to be Hausdorff-continuous, to take bounded, convex and closed values, and to satisfy weak continuity with respect to one variable.

math.AP↗
arXiv2026

Avscon the Schur topology

The aim of the course is to lead to an understanding of homogenisation processes in an operator-theoretic sense. In fact, using solely operator-theoretic means not referring to the particular form of the coefficients, we will identify an operator topology on the level of coefficients that will fully capture the convergence involved in the context of homogenisation. One upshot of this perspective will be that we will obtain homogenisation results for time-dependent partial differential equations (almost) for free.

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arXiv2026

Exponential Stability for Maxwell-type Systems Revisited

Considering a two-by-two block operator matrix system of Maxwell type, we present an elementary way of deducing exponential stability under minimal smoothness (and boundedness) requirements of the underlying domains when applications are concerned. The approach is based on resolvent estimates using block operator matrices.

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arXiv2026

Block operator matrix techniques for stability properties of hyperbolic equations

Inspired by recent developments in the theory of stability results in the context of certain wave type phenomena, we discuss abstract damped hyperbolic type equations given in a block operator matrix form with regards to asymptotic behaviour of their solutions. Under mild conditions on the operators involved we provide criteria establishing strong or semi-uniform stability. In the particular case of Maxwell's equations, these criteria are implied under mild regularity conditions of the underlying domain causing spatial derivative operators satisfy certain compact embedding conditions and rather minimal assumptions on the damping conductivity. These assumptions improve on both regularity as well as on the structural requirements for the conductivity previously available in the literature.

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arXiv2026

The Closed Range Property and Gaffney's Inequality of the De Rham Complex in Unbounded Domains

The classical Poincar\'e estimate establishes closedness of the range of the gradient in unweighted $L^2(\Omega)$-spaces as long as $\Omega\subseteq\mathbb{R}^3$ is contained in a slab, that is, $\Omega$ is bounded in one direction. Here, as a main observation, we provide closed range results for the $\operatorname{rot}$-operator, if (and only if) $\Omega$ is bounded in two directions. Along the way, we characterise closed range results for all the differential operators of the primal and dual de Rham complex in terms of directions of boundedness of the underlying domain. As a main application, one obtains the existence of a spectral gap near the $0$ of the Maxwell operator allowing for exponential stability results for solutions of Maxwell's equations with sufficient damping in the conductivity. Our results are based on the validity of Gaffney's (in)equality and the transition of the same to unbounded (simple) domains as well as on the stability of closed range results under bi-Lipschitz regular transformations. The latter technique is well-known and detailed in the appendix; for the results concerning Gaffney's estimate, we shall provide accessible, simple proofs using mere standard results. Moreover, we shall present non-trivial examples and a closed range result for $\operatorname{rot}$ with mixed boundary conditions on a set bounded in one direction only.

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arXiv2026

Characterisation of homogenisation for nonlocal diffusion by local topologies

We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical $H$-convergence covering the local behaviour of the fractional divergence form problem and weak-$\ast$ convergence on the complement caused by the nonlocality of the differential operators. The results are further described in the light of nonlocal $H$-convergence as introduced in [Waurick, Calc Var PDEs, 57, 2018] and certain Schur topologies. Applications to symmetric coefficients and a homogenisation problem for a fractional heat type equation are provided.

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arXiv2025

Spatial Approximation for Evolutionary Equations

We consider evolutionary equations as introduced by R.\ Picard in 2009 and develop a general theory for approximation which can be seen as a theoretical foundation for numerical analysis for evolutionary equations. To demonstrate the approximation result, we apply it to a spatial discretisation of the heat equation using spectral methods.

math.FA↗
arXiv2025

Evolutionary equations with state-dependent delay

We extend a contraction mapping argument for ordinary state-dependent delay differential equations to evolutionary partial differential equations in the sense of R. Picard, that is, to equations of the form $\bigl(\partial_{t} M(\partial_{t}) + A\bigr) u(t) = F\bigl(t,u_{(t)}\bigr)$, where $A$ is an $\mathrm{m}$-accretive (unbounded) linear operator and $M$ is a material law. We establish local well-posedness (in the sense of weak solutions) of generalized initial value problems that stem from a distributional formulation. We require prehistories in $H^{1}$ with bounded derivative, a regularity increasing right-hand side and a consistency condition. We showcase the viability of our results by applying them to classical examples (heat, wave and Maxwell's equations), examples from semigroup theory, port-Hamiltonian systems, as well as equations featuring fractional derivatives and convolutions (in time) with bounded operators.

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arXiv2024

A quick guide to ordinary state-dependent delay differential equations

We review $H^{1}$-well-posedness for initial value problems of ordinary differential equations with state-dependent right-hand side. We streamline known approaches to infer existence and uniqueness of solutions for small times given a Lipschitz-continuous prehistory. The paramount feature is a reduction of the differential equation to a fixed point problem that admits a unique solution appealing to the contraction mapping principle. The use of exponentially weighted Sobolev spaces in this endeavor proves to be as powerful as for ordinary differential equations without delay. Our result includes a blow-up criterium for global existence of solutions. The discussion of well-posedness is concluded by new results covering continuous dependence on initial prehistories and on the right-hand sides.

math.CA↗
arXiv2024

A universal example for quantitative semi-uniform stability

We characterise quantitative semi-uniform stability for $C_0$-semigroups arising from port-Hamiltonian systems, complementing recent works on exponential and strong stability. With the result, we present a simple universal example class of port-Hamiltonian $C_0$-semigroups exhibiting arbitrary decay rates slower than $t^{-1/2}$. The latter is based on results from the theory of Diophantine approximation, as the decay rates will be strongly related to the approximation properties of irrational numbers by rationals obtained from cut-offs of continued fraction expansions.

math.AP↗
arXiv2024

Homogenisation for Maxwell and Friends

We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.

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arXiv2024

Fibre homogenisation for time-dependent problems

In this article we provide a method for establishing operator-type error estimates between solutions to rapidly oscillating evolutionary equations and their homogenised counter parts. This method is exemplified by applications to the wave, heat and finally thermoelastic evolutionary systems.

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arXiv2024

A simple way to well-posedness in $H^{1}$ of a delay differential equation from cell biology

We present an application of recent well-posedness results in the theory of delay differential equations for ordinary differential equations arXiv:2308.04730 to a generalized population model for stem cell maturation. The weak approach using Sobolev-spaces we take allows for a larger class of initial prehistories and makes checking the requirements for well-posedness of such a model considerably easier compared to previous approaches. In fact the present approach is a possible means to guarantee that the solution manifold is not empty, which is a necessary requirement for a $C^{1}$-approach to work.

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arXiv2023

Adjoints of sums of m-accretive operators and applications to non-autonomous evolutionary equations

We provide certain compatibility conditions for m-accretive operators such that the adjoint of the sum is given by the closure of the sum of the respective adjoint. We revisit the proof of well-posedness of the abstract class of partial differential-algebraic equations known as evolutionary equations. We show that the general mechanism provided here can be applied to establish well-posedness for non-autonomous evolutionary equations with $L_{\infty}$-coefficients thus not only generalising known results but opening up new directions other methods such as evolution families have a hard time to come by.

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