Wolf von Wahl

On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations

We study regularity properties of weak solutions of the eqns. of Navier-Stokes which are in L∞((O,T),Ln(Ω)) or in LP((O,T),Ln(Ω)) for some p≧2. We prove also that L∞((O,T), Ln(Ω)) is a uniqueness class for weak solutions. Moreover we give a generalization of Serrins uniqueness result.

Time dependent nonlinear Schrödinger equations

We prove the existence of global classical solutions of the Cauchy-problem for nonlinear Schrödinger equations u′+iAu+f(|u|2)u or u′+Au+if(|u|2)u respectively. We need various growth conditions on the nonlinearity f and some restrictions on the admissible space dimension n.

Estimating ∇u in terms of div u, curl u, either (ν, u) or ν×u and the topology

In the present paper we prove Cα-estimates for ∇u using components of boundary values of u, div u, curl u and quantities given by components of boundary values of u as well as boundary values of elements belonging to de Rhams cohomology modules. The vector field u is defined on a bounded set G¯⊂ℝ3, meanwhile the cohomology group will be defined with regard to ℝ3−G. Our inequalities turn out to be a priori estimates concerning well-known boundary value problems for vector fields.

The Eckhaus criterion for convection roll solutions of the Oberbeck-Boussinesq equations

Abstract This paper is concerned with the stability problem of the convection roll solutions of the Oberbeck-Boussinesq equations. We consider the stability of the roll solutions with respect to the two-dimensional disturbances. The Eckhaus criterion for the stability and instability of the roll solutions is derived in a mathematically rigorous way.

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

SummaryWe perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.

Introduction to the Cosserat problem

The study of the Cosserat spectrum started more than 100 years ago with a series of papers [2]–[10] published between 1898 and 1901 by the French scientists Eugène and François Cosserat. Their motivation was to expand the solutions of certain basic problems of static elasticity into eigenvectors. Let BR := {x ∈ Rn : |x| < R} for n ≥ 2. In case n = 3 they tried to solve the following boundary value problem: If u0 = (u01, u02, u03) ∈ [ C∞(BR) ∩ C(B̄R) ]3 satisfying u0 = 0 in BR is given, find for σ ∈ R a solution of u + σ∇ div u = 0 in BR, u |∂BR= u0 |∂BR . (0.1)

Regularity of weak solutions of semilinear parabolic systems of arbitrary order

Letu be a weak solution of the initial boundary value problem for the semilinear parabolic system of order 2m:u′(t)+Au(t)+f(t,.,u,..., ▽mu)=0. Letf satisfy controllable growth conditions. Thenu is smooth.This result is proved by a kind of continuity method, where the timet is the parameter of continuity.

On parabolic systems with discontinuous nonlinearities

We consider parabolic systems over (O,T)×Ω with bounded but discontinuous nonline-arities. Here A1,A2 are positive elliptic operators of order 2 m with continuous coefficients, f is a bounded function having a jump in u=1, and g1, g2 are Lipschitz continuous and bounded. We prescribe Dirichlet boundary conditions and the initial values u(O)=φ1, v(O)=φ2. Essentially under the condition f(u o) =O we then prove the existence of global solutions which are almost regular everywhere, i.e.: u′ , v′ ϵ Lp(Ω)), u,v ϵ L p ((O,T), H2m,p(Ω)) for some large p and for all T > O and consequently λ2m−1u, λ2m−1v are Holder continuous in t, x over . Our proof is based on the construction of a putative solution (u,v) by approximation and the study of the set u =uo. Although we have in mind applications to second order equations with A1 =A2=−λ we purposely work within a more general framework; thereby we want to show that we need neither the maximum principle nor any monotonicity properties of the nonlinear part.

Asymptotic stability of higher order norms in terms of asymptotic energy stability for viscous incompressible fluid flows heated from below

We consider a steady viscous incompressible fluid flow in an infinite layer heated from below. The steady flow is assumed to be periodic with respect to the plane variables. If this flow turns out to be asymptotically energy-stable with respect to a particular disturbance then it is also asymptotically stable in higher order norms with respect to the same perturbation. No smallness of the initial values is needed.

Remarks on Lines of reversibility for Poincaré´s centre problem

where p, q are polynomials whose terms of lowest order are of degree at least two. A well-known sufficient condition, due to Poincaré, for the origin to be a centre is that the system be reversible with respect to a line L, which passes through the origin, i.e. that the system be invariant under a reflection in the line L, and under a simultaneous reversal of the independent variable t. Thus system (1.1) is reversible with respect to the line x = 0 if and only if it is invariant under the transformation (x, y, t) → (−x, y,−t), i.e. if and only if q(−x, y) = q(x, y) and p(−x, y) = −p(x, y). Thus q contains only even powers of x and p only odd ones. Reversibility with respect to y = 0 is thus equivalent to q(x,−y) = −q(x, y), p(x,−y) = p(x, y), i.e. q contains only odd powers of y and p only even ones. As for a general line L one can apply a rotation which transforms L into the line x = 0 or y = 0 and a criterion for reversibility may then readily be attained [1]. Collins, by using tensor-calculus, derives in [2] a necessary and sufficient condition for the existence of such a line without involving its unknown equation.