Degenerate Diffusions

The use of parabolic double obstacles problems for approximating curvature dependent phase boundary motion is reviewed. It is shown that such problems arise naturally in multi-component diffusion with capillarity. Formal matched asymptotic expansions are employed to show that phase field models with order parameter solving an obstacle problem approximate curvature dependent phase boundary motion. Numerical simulations of surfaces evolving according to their mean curvature are presented.

This note is to announce some new results and techniques in the theory of singular parabolic equations of the type

In this short note, we prove a BMO bound on eigenfunctions of the Laplacian on Riemannian manifold. This bound is closely related to a conjecture on the Hausdorff measure of the nodal set.

We consider the Cauchy problem for a one-dimensional quasilinear degenerate heat equation with source. The property of monotone in time behavior of the weak solution at a fixed spatial point x?ϵ R is studied. It is shown that the conditions of a such a behavior depend on “a nonlinear interaction” between the nonlinear heat operator and the source of energy considered. There are two different cases: (i) the solution is monotone in time for any x0 which is far enough from the initial support and (ii) the solution is monotone in time if it becomes large enough. In a general situation the monotone in time behavior at a given point x — x0 is proved to depend on the shape of the initial function in some neighborhood of x = x0. Proofs are based on the method of intersection comparison of the solution and the continuous set of stationary solutions of the same equation.

Let Ω0 be a compact region of the three-dimensional euclidean space R3. Assume that Ω:= R3 — Ω0 is filled with a viscous incompressible fluid whose motion m is governed by the Navier-Stokes equations.

We discuss on recent results concerning the asymptotics near blow-up of nonnega-tive solutions of where f (u)=u ^p with p > 1 or f (u) = eu, and u 0(x) is continuous, nonnegative and bounded. AMS(MOS) subject classifications. 35B40, 35K55, 35K57

The purpose of this expository paper is to describe a new method, introduced in a series of papers [LNT], [NT1,2], [NPT] and [J], in handling “spikes” (or “point-condensation” phenomena) for singularly perturbed semilinear elliptic equations of the form where \( \Delta = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}}}{{\partial x_i^2}}} \) is the Laplace operator in R ⁿ , and ε is a small positive number.

This paper is a sequel to our paper [OU] where we investigated questions concerning solvability and asymptotic behavior of solutions to the mean curvature evolution problem where Ω is a bounded domain in R ⁿ , n ≥ 2, with C^∞ boundary ∂Ω, H is the mean curvature operator.

We shall be concerned with continuation and limit behavior as r → ∞ for solutions of the quasi-variational system

In this paper, we shall describe some recent results, ([6,7]), concerning the existence of smooth, globally defined static solutions of the Einstein /Yang-Mills (EYM) equations. The EYM equations extend Einstein’s celebrated gravitational field equations, to include other “force” fields: electromagnet ism, and the weak and strong nuclear force fields. It turns out that solutions of such coupled systems of equations, in n = 4 space-time dimensions, have some very interesting mathematical and physical properties. Thus for the non-abelian gauge group G = SU (2), we can prove the existence of infinitely many non-singular globally defined solutions - a result that fails to hold if G = U (1), (the coupling of gravity to electromagnetism). Since the (classical) Yang-Mills equations with G = SU (2) correspond to the weak nuclear force, our result indicates that coupling gravity to the weak nuclear force can prevent singularity formation in space-time. Furthermore, these solutions give rise to constants which correspond to masses. In other words, numbers representing masses are a direct consequence of our theory, and need not be prescribed in an ad-hoc manner - as is usually done in physical theories.

For a given domain Ω⊂ R n, we consider the variational problem of minimizing the L1-norm of the gradient on Ωof a function with prescribed continuous boundary values. Under certain weak conditions on the boundary of the domain Ω, it is shown that the BV solution is continuous and unique.

In this lecture I will try to present a certain view of the progress achieved in recent years on the subject of asymptotic behaviour of nonlinear parabolic equations. This is an enormously rich field since we can play with the domain, the initial data, the boundary data and the form of the equation. Therefore, a selection of the topics to treat is necessary. Our interest will be centered around the stabilization of solutions to an equilibrium state, with special emphasis on the calculation of the characteristic exponents, namely the decay rate and the expansion rate.