The Principal Floquet Bundle and Exponential Separation for Linear Parabolic Equations

We consider linear nonautonomous second order parabolic equations on bounded domains subject to Dirichlet boundary condition. Under mild regularity assumptions on the coefficients and the domain, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. Our main theorem extends in a natural way standard results on principal eigenvalues and eigenfunctions of elliptic and time-periodic parabolic equations. Similar theorems were earlier available only for smooth domains and coefficients. As a corollary of our main result, we obtain the uniqueness of positive entire solutions of the equations in     

On the non-classical problem of thermal conduction on anisotropic layers in R3

We propose a solution methodology for boundary problems of parabolic and hyperbolic thermal conduction on anisotropic layers in R³. We study the wave nature of heat transfer with pulse thermal effects in bounded bodies with cavity. We compare the solutions of the parabolic and hyperbolic equations of thermal conduction and we show that the assumption on the wave nature of energy transfer is justified under the conditions for high-speed processes.     

Infinite-Dimensional Hyperbolic Sets and Spatio-Temporal Chaos in Reaction Diffusion Systems in $${\mathbb{R}^{n}}$$

The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space–time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space–time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space–time chaos. Obviously, hyperbolicity is lost in this step. Research partially supported by the INTAS project Attractors for Equations of Mathematical Physics, by CRDF and by the Alexander von Humboldt–Stiftung.     

Diffusion-Induced Blowup in a Nonlinear Parabolic System

A two-component semilinear parabolic system on a bounded domain with Neumann boundary conditions is studied. It is shown that for a certain kind of nonlinearity, the blowup of solutions may occur when the diffusion coefficients are not equal, though the corresponding ODE possesses a globally stable equilibrium.     

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.     

Existence of Weak Solutions for the Motion of Rigid Bodies in a Viscous Fluid

. We study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of with Dirichlet boundary conditions. By introducing an appropriate weak formulation for the complete problem, we prove existence of solutions for initial velocities in . In the absence of collisions, solutions exist for all time in dimension 2, whereas in dimension 3 the lifespan of solutions is infinite only for small enough data. (Accepted June 10, 1998)     

Existence and Uniqueness Theorems for the Two-Dimensional Ericksen–Leslie System

In this paper we study the two dimensional Ericksen–Leslie equations for the nematodynamics of liquid crystals if the moment of inertia of the molecules does not vanish. We prove short time existence and uniqueness of strong solutions for the initial value problem in two situations: the space-periodic problem and the case of a bounded domain with spatial Dirichlet boundary conditions on the Eulerian velocity and the cross product of the director field with its time derivative. We also show that the speed of propagation of the director field is finite and give an upper bound for it.     

A Note on Diffusion-Induced Blow-up

The goal of paper is to give a simpler proof and some extensions of a result of Weinberger [3] concerning diffusion induced blow-up. The result states that, for certain systems of two parabolic equations with equal diffusion and homogeneous Neumann boundary conditions, some blowing-up solutions exist, although the corresponding system of ODE’s has only global bounded solutions. To Professor Pavol Brunovsky, on the occasion of his 70th birthday.     

Solution of three-dimensional viscous incompressible flows by a multi-grid method

The three-dimensional Navier-Stokes equations for viscous incompressible fluids are discretized on staggered or non-staggered grids. The system of finite-difference equations is solved by a multi-grid method. The method and some possible sources of difficulties and their remedies are described. The numerical algorithm has been applied to the computations of flows in ducts for a range of Reynolds numbers up to 2000. As outflow boundary conditions, either the fully developed flow profile (Dirichlet condition) or parabolic conditions have been applied. The multi-grid method has a fast rate of convergence (with both types of boundary conditions), and it is not sensitive to the number of mesh points and the Reynolds number. The numerical solution, using parabolic boundary conditions, is insensitive to the location of the outflow boundary, even for large Reynolds numbers, in contrast to the solution with Dirichlet boundary conditions.     

A generalization of the method of averaging for systems with two time scales

In this paper we shall consider systems of the form x = ? f(t, ?t, x, y, ?), y = g(t,?t, x, y,?), where x and y are vectors of finite dimensions, f and g are assumed to be bounded for all t, and ? is a real parameter. Sufficient conditions are obtained for the existence of certain solutions which are bounded for all t. These solutions are shown to approach special solutions of a derived simpler averaged system of equations as ? → 0. Moreover, it is shown that there exists only one such bounded solution in the neighborhood of each special solution. In the special case when y is not present, it is shown that if a special solution is stable, solutions starting in nonlocal neighborhoods of this special solution approach the bounded solutions adjacent to it as t → ∞. These results generalize most of the existing work for systems of the type discussed here. Finally, we employ our results to study some problems of physical importance.     

Radial Symmetry of Overdetermined Boundary-Value Problems in Exterior Domains

In this paper, we extend a classical result by J. Serrin [15] to exterior domains , where Ω is a bounded domain. We prove, under some hypotheses on f, that if there exists a solution of satisfying the overdetermined boundary conditions that and u are constant on , and such that , then the domain Ω is a ball. Under different assumptions on f, this result has been obtained by W. Reichel in [13]. The main result here covers new cases like with . When Ω is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet condition that u is constant on . Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical position. (Accepted May 30, 1997)     

Blow-up rate and profile for a class of quasilinear parabolic system

This paper deals with positive solutions to a class of nonlocal and degenerate quasilinear parabolic system with null Dirichlet boundary conditions. The blow-up rate and blow-up profile are gained if the parameters and the initial data satisfy some conditions.     

Long-Time Asymptotics of a Multiscale Model for Polymeric Fluid Flows

In this paper, we investigate the long-time behavior of some micro-macro models for polymeric fluids (Hookean model and FENE model), in various settings (shear flow, general bounded domain with homogeneous Dirichlet boundary conditions on the velocity, general bounded domain with non-homogeneous Dirichlet boundary conditions on the velocity). We use both probabilistic approaches (coupling methods) and analytic approaches (entropy methods).     

Diffusional growth of cloud particles: existence and uniqueness of solutions

Diffusional growth of cloud particles is commonly described by a coupled system of parabolic equations and ordinary differential equations. The Dirichlet boundary condition for the parabolic equation is obtained from the solution of the ordinary differential equations, but this solution itself depends on the solution of the parabolic equations. We first present the governing equations describing diffusional growth of cloud particles. In a second step, we consider a simplified model problem, motivated by the diffusional growth equations. The main difference between the simplified model problem and the diffusional growth equations consists in neglecting the dependence of the domain for the parabolic equations on the solution. For the model problem, we show unique solvability using a fixed point method. Finally, we discuss application of the main result for the model problem to the diffusional growth equations and illustrate these equations with the help of a numerical solution.     

Résumé and remarks on the open boundary condition minisymposium

The incompressible Navier-Stokes equations—and their thermal convection and stratified flow analogue, the Boussinesq equations—possess solutions in bounded domains only when appropriate/legitimate boundary conditions (BCs) are appended at all points on the domain boundary. When the boundary—or, more commonly, a portion of it—is not endowed with a Dirichlet BC, we are faced with selecting what are called open boundary conditions (OBCs), because the fluid may presumably enter or leave the domain through such boundaries. The two minisymposia on OBCs that are summarized in this paper had the objective of finding the best OBCs for a small subset of two-dimensional test problems. This objective, which of course is not really well-defined, was not met (we believe), but the contributions obtained probably raised many more questions/issues than were resolved—notable among them being the advent of a new class of OBCs that we call FBCs (fuzzy boundary conditions).     

Geometry of Heteroclinic Cascades in Scalar Parabolic Differential Equations

We investigate the geometrical properties of the attractor for semilinear scalar parabolic PDEs on a bounded interval with Neumann boundary conditions. Using the nodal properties of the stationary solutions which are determined by an ordinary boundary value problem, we obtain crucial information about the long-time behavior for the full PDE. Especially, we prove an exact criterion for the intersection of strong-stable and strong-unstable manifolds in the finite dimensional Morse-Smale flow on the attractor.     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Effective Boundary Conditions Resulting from Anisotropic and Optimally Aligned Coatings: The Two Dimensional Case

We discuss a thermally conducting body insulated by a thin anisotropically conducting coating. The coating is “optimally aligned” in the sense that the normal vector inside the coating is always an eigenvector of the thermal tensor. We study the effects of the coating by investigating the limiting behavior of solutions u of the heat equation with either Dirichlet or Neumann boundary conditions imposed on the outer boundary of the coating, as the thickness of the coating shrinks to zero. In the two-dimensional case, we find the complete list of “effective boundary conditions” satisfied by the limit of u on the boundary of the uncoated body. This list contains not only the usual Dirichlet, Neumann and Robin boundary conditions, but also some new and even nonlocal ones involving the Dirichlet-to-Neumann mapping and the Hilbert transform on the circle. We also prove that u converges to its limit in various norms that include the L ², the Sobolev and the Hölder ones. During the course of this study, we establish a Schauder theory for the regularity of weak solutions of general second order parabolic equations near an interface where the “transmission condition” is satisfied.     

Measure-valued solutions of scalar conservation laws with boundary conditions

We define a solution concept for measure-valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions. This result may be used to prove convergence, towards the unique solution, for approximate solutions which are uniformly bounded in L , weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.     

Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations

The stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded intervalJ of the real line with homogeneous Dirichlet boundary conditions is studied. It is shown that it is possible to have stable stationary solutions which change sign once onJ in the case of constant diffusion when the reaction term does not depend explicitly on the space variable. The problem of the possible types of stable solutions that may exist is considered. It is also shown that Matano's result on the lap-number is still true in the case of nonlocal problems.On leave from IST, Lisbon, Portugal.