Jürgen Saal

Nonlinear Partial Differential Equations

Nonlinear partial differential equations (PDE’s) play a central role in the modelling of a great number of phenomena, ranging from Theoretical Physics, Astrophysics and Chemistry to Economy, Medicine and Population Dynamics. Among the phenomena encountered, the diffusion processes play a fundamental role. In the last 25 years a great deal of work has been devoted to semi-linear equations. In this type of equations, the interaction between a linear partial differential operator and the superlinear reaction term (source or absorption) can be understood, at least in part, thanks to the linear theory. One of the main observations, valid in the most interesting cases is the existence of critical exponents, for example the Fujita exponents for nonlinear heat equation, the Pohozaev exponent, the Sobolev exponent. In general, the first results (blow-up, global estimates, decay estimate), were proved up to a critical exponent by more or less easy applications of linear energy estimates, linked to ODE techniques. Then the study of what happens if the exponent is critical or even supercritical involves a very delicate analysis, often based upon very sharp linear estimates or even, in some cases a completely new approach. The main areas of research in the current proposal include the description of singular phenomenon: blow-up, singularities, problems with singular measure data in a large class of reaction diffusion equations, with quasi-linear or fully nonlinear diffusion and strong reaction.

The Stokes Operator with Robin Boundary Conditions in Solenoidal Subspaces of L 1(ℝ n +) and L ∞(ℝ n +)

We prove that the Stokes operator with Robin boundary conditions is the generator of a bounded holomorphic semigroup on , which is even strongly continuous on the space BUCσ(ℝ n +). Contrary to that result it is also proved that there exists no Stokes semigroup on , except if we assume the special case of Neumann boundary conditions. Nevertheless, we also obtain gradient estimates for the solution of the Stokes equations in for all types of Robin boundary conditions.

R-sectoriality of Cylindrical Boundary Value Problems

We prove \( \mathcal{R} \)-sectoriality or, equivalently, L p -maximal regularity for a class of operators on cylindrical domains of the form \( \mathbb{R}^{n-k}\times V \), where \( V \subset \mathbb{R}^{k} \) is a domain with compact boundary, \( \mathbb{R}^{k} \), or a half-space. Instead of extensive localization procedures, we present an elegant approach via operatorvalued multiplier theory which takes advantage of the cylindrical shape of both, the domain and the operator.

Self-Similar Solutions for Various Equations

We first present for the porous medium equation, a typical nonlinear degenerate diffusion equation, that its (forward) self-similar solution well describes asymptotic behavior of solutions, as is observed for the heat equation, without proof. We next explain that it is important to classify backward self-similar solutions in order to analyze behavior of solutions near singularities for the axisymmetric mean curvature flow equation as an example. In what follows, a self-similar solution is regarded as a stationary solution of the equation written with similarity variables. Convergence behavior of a solution of the equation to its stationary corresponds to the asymptotic behavior of the solution of the original equation near singularities. We give an outline of the proof of convergence and mention that a monotonicity formula plays a key role. Moreover, we give a simple proof of uniqueness of the stationary solutions, i.e., the backward self-similar solutions of the original equation. The proof is simpler and easier than that in the literature. We remark that the method using similarity variables is applicable, to some extent, to other diffusion equations such as semilinear heat equations and harmonic map flow equations. Finally, we note that the existence of forward self-similar solutions has also been proved for nonlinear equations of nondiffusion type.

Analysis of a Living Fluid Continuum Model

Generalized Navier-Stokes equations which were proposed recently to describe active turbulence in living fluids are analyzed rigorously. Results on wellposedness and stability in the

Convergence Theorems in the Theory of Integration

L^2(\mathbb{R}^n)

Behavior Near Time Infinity of Solutions of the Vorticity Equations

-setting are derived. Due to the presence of a Swift-Hohenberg term global wellposedness in a strong setting for arbitrary initial data in

Behavior Near Time Infinity of Solutions of the Heat Equation

L^2_\sigma(\mathbb{R}^n)

Various Properties of Solutions of the Heat Equation

is available. Based on the existence of global strong solutions, results on linear and nonlinear (in-) stability for the disordered steady state and the manifold of ordered polar steady states are derived, depending on the involved parameters.

Existence of analytic solutions for the classical Stefan problem

This section gives a summary of some elementary facts used frequently throughout this book, and can be regarded as an appendix. In particular, we consider sufficient conditions for the interchange of integration and limit operations. In detail, we discuss a result on uniform convergence, the dominated convergence theorem, the bounded convergence theorem, Fatou’s lemma, and the monotone convergence theorem from the points of view of both Lebesgue integration theory and Riemann integration theory. Note that these are well-known results; hence we will be brief in details. For the proof of the monotone convergence theorem and Fubini’s theorem we merely refer to the appropriate literature.