Michael Dreher

Ill-posed problems in thermomechanics

Abstract Several thermomechanical models have been proposed from a heuristic point of view. A mathematical analysis should help to clarify the applicability of these models, among those recent thermal or viscoelastic models. Single-phase-lag and dual-phase-lag heat conduction models can be interpreted as formal expansions of delay equations. The delay equations are shown to be ill-posed, as are the formal expansions of higher order — in contrast to lower-order expansions leading to Fourier’s or Cattaneo’s law. The ill-posedness is proved, showing the lack of continuous dependence on the data, and thus showing that these models (delay or higher-order expansion ones) are highly explosive. In this note we shall present conditions for when this happens.

The viscous model of quantum hydrodynamics in several dimensions

We investigate the viscous model of quantum hydrodynamics in one and higher space dimensions. Exploiting the entropy dissipation method, we prove the exponential decay to the thermal equilibrium state in one, two, and three dimensions, provided that the domain is a box. Further, we show the local in time existence of a solution in the one-dimensional case; and in the case of higher dimensions under the assumption of periodic boundary conditions. Finally, we prove the global existence in a one-dimensional setting under additional assumptions.

Quantum Semiconductor Models

We give an overview of analytic investigations of quantum semiconductor models, where we focus our attention on two classes of models: quantum drift diffusion models, and quantum hydrodynamic models. The key feature of those models is a quantum interaction term which introduces a perturbation term with higher-order derivatives into a system which otherwise might be seen as a fluid dynamic system. After a discussion of the modeling, we present the quantum drift diffusion model in detail, discuss various versions of this model, list typical questions and the tools how to answer them, and we give an account of the state-of-the-art of concerning this model. Then we discuss the quantum hydrodynamic model, which figures as an application of the theory of mixed-order parameter-elliptic systems in the sense of Douglis, Nirenberg, and Volevich. For various versions of this model, we give a unified proof of the local existence of classical solutions. Furthermore, we present new results on the existence as well as the exponential stability of steady states, with explicit description of the decay rate.

Viscous quantum hydrodynamics and parameter-elliptic systems

The viscous quantum hydrodynamic model derived for semiconductor simulation is studied in this paper. The principal part of the vQHD system constitutes a parameter{elliptic operator provided that boundary conditions satisfying the Shapiro{Lopatinskii criterion are specied. We classify admissible boundary conditions and show that this principal part generates an analytic semigroup, from which we then obtain the local in time well{posedness. Furthermore, the exponential stability of zero current and large current steady states is proved, without any kind of subsonic condition. The decay rate is given explicitly.

Analysis of a Population Model with Strong Cross-Diffusion in Unbounded Domains

We study a parabolic population model in the full space and prove the global in time existence of a weak solution. This model consists of two strongly coupled diffusion equations describing the population densities of two competing species. The system features intrinsic growth, inter- and intra-specific competition of the species, as well as diffusion, cross-diffusion and self-diffusion, and drift terms related to varying environment quality. The cross-diffusion terms can be large, making the system non-parabolic for large initial data. For the first time, solutions in unbounded domains are studied. The method of our proof is a combination of a time semi-discretization, a transformation of the system guaranteeing the positivity of the solution, a special entropy symmetrizing the system, and compactness arguments. 2000 Mathematics Subject Classification: 35K55 (primary), 92D25, 35D05 (secondary).

Energy estimates for weakly hyperbolic systems of the first order

For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp.

Necessary conditions for the well-posedness of Schrödinger type equations in Gevrey spaces

We discuss evolution operators of Schrodinger type which have a non-self-adjoint lower order term and give a necessary condition for the Cauchy problem to such operators to be well-posed in Gevrey spaces. Under an additional assumption, this necessary condition is sharp.

Sharp Energy Estimates for a Class of Weakly Hyperbolic Operators

The intention of this article is twofold: First, we survey our results from [20, 18] about energy estimates for the Cauchy problem for weakly hyperbolic operators with finite time degeneracy at time t = 0. Then, in a second part, we show that these energy estimates are sharp for a wide range of examples. In particular, for these examples we precisely determine the loss of regularity that occurs in passing from the Cauchy data at t = 0 to the solutions.

Decay Estimates of Solutions to Wave Equations in Conical Sets

We consider the wave equation in an unbounded conical domain, with initial conditions and boundary conditions of Dirichlet or Neumann type. We give a uniform decay estimate of the solution in terms of weighted Sobolev norms of the initial data. The decay rate is the same as in the full space case.

Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations

The purpose of this paper is to investigate weakly hyperbolic equations with degeneracies in the space and time variables. These degeneracies as well as the sharp Levi conditions of C ∞ type are formulated by means of certain weight functions. For Cauchy problems to such quasi-linear weakly hyperbolic equations, the following subjects are studied: local existence of solutions in Sobolev spaces and C ∞, a blow-up criterion, domains of dependence, and C ∞ regularity. The main tools are the transformation of the higher-order equation to a first-order system, a calculus for pseudodifferential operators with non-smooth symbols, and a generalization of Gronwall’s lemma to differential inequalities with a singular coefficient.