On Leray's Self‐Similar Solutions of the Navier‐Stokes Equations Satisfying Local Energy Estimates

This paper proves that Leray's self‐similar solutions of the three‐dimensional Navier‐Stokes equations must be trivial under very general assumptions, for example, if they satisfy local energy estimates. (Accepted April 7, 1997)     

Exact Solution of the Wave‐Resistance Problem for a Submerged Cylinder. I. Linearized Theory

. We consider the problem of finding a holomorphic function in a strip with a cut ${\cal A}= \{(x,y) : \, x\in\RE,\,\,0 satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for $|x|\to\infty. We consider the problem of finding a holomorphic function in a strip with a cut satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for . A special solution can be selected by fixing the value of the circulation around the cut. The problem is obtained by linearization of the equations of the wave‐resistance problem for a “slender” cylinder submerged in a heavy fluid and moving at uniform speed in the direction orthogonal to its generators. The results obtained, besides their own interest, are a crucial step for the resolution of the non‐linear problem. (Accepted October 14, 1998)     

On the Initial‐Boundary‐Value Problem for the Linearized Equations of Magnetohydrodynamics

We study the initial‐boundary‐value problem for the linearized equations of ideal magnetohydrodynamics defined on a bounded domain whose boundary is a regular magnetic surface. We show that it is well‐posed in and that the loss of regularity of solutions always occurs for some initial data in (Accepted October 23, 1997)     

Divergence‐Measure Fields and Hyperbolic Conservation Laws

. We analyze a class of vector fields, called divergence‐measure fields. We establish the Gauss‐Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of fields. Then we apply this theory to analyze entropy solutions of initial‐boundary‐value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class of systems with affine characteristic hypersurfaces. The analysis in also extends to . (Accepted July 16, 1998)     

Nonlinearly Elastic Shell Models: A Formal Asymptotic Approach I. The Membrane Model

We consider a family of three‐dimensional shells with the same middle surface, all composed of the same nonlinearly elastic Saint Venant‐Kirchhoff material. Using the method of asymptotic expansions with the thickness as the “small” parameter, and making specific assumptions on the applied forces, the geometry of the middle surface, and the kinematic boundary conditions, we show how a “limiting”, “large‐deformation” two‐dimensional model can be identified in this fashion. By linearization, this nonlinear membrane model reduces to the linear membrane model. (Accepted January 13, 1997)     

Existence of Minimizers for a Variational Problem in Two‐Dimensional Nonlinear Magnetoelasticity

We prove the existence of energy‐minimizing configurations for a two‐dimensional, variational model of magnetoelastic materials capable of large deformations. The model is based on an energy functional which is the sum of the nonlocal self‐energy (the energy stored in the magnetic field generated by the body, and permeating the whole ambient space) and of the local anisotropy energy, which is not weakly lower semicontinuous. A further feature of the model is the presence of a non‐convex constraint on one of the unknowns, the magnetization, which is a unit vector field. (Accepted November 20, 1997)     

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globally in time, the velocity distribution f(v,t) of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field E. The density f satisfies a Boltzmann-type kinetic equation containing a fully nonlinear electron‐electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the L ¹ distance between f and a certain time-dependent Maxwellian stays small uniformly in t. Moreover, the mean and variance of this time‐dependent Maxwellian satisfy a coupled set of nonlinear ordinary differential equations that constitute the “hydrodynamical” equations for this kinetic system. This remains true even when these ordinary differential equations have non‐unique equilibria, thus proving the existence of multiple stable stationary solutions for the full kinetic model. Our approach relies on scale‐independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globally in time. (Accepted September 3, 1996)     

Young‐Measure Solutions for a Viscoelastically Damped Wave Equation with Nonmonotone Stress‐Strain Relation

We study the viscoelastically damped wave equation with a nonmonotone stress‐strain relation σ. This system describes the dynamics of phase transitions, which is closely related to the creation of microstructures. In order to analyze the dynamic behavior of microstructures we consider highly oscillatory initial states. Two questions are addressed in this work: How do oscillations propagate in space and time? What can be said about the long‐time behavior? An appropriate tool to deal with oscillations are Young measures. They describe the local distribution or one‐point statistics of a sequence of fast fluctuating functions. We demonstrate that highly oscillatory initial states generate in a unique fashion an evolution in the space of Young measures and we derive the determining equations. Further on we prove a generalized dissipation identity for Young‐measure solutions. As a consequence, it is shown that every low‐energy solution converges to a Young‐measure equilibrium as t→∞. This is a generalization of G. Friesecke's & J. B. McLeod's [FM96] convergence result for classical solutions to the case of Young‐measure solutions. (Accepted November 12, 1997)     

Viscosity Solutions of a Degenerate Parabolic-Elliptic System Arising in the Mean-Field Theory of Superconductivity

In a Type‐II superconductor the magnetic field penetrates the superconducting body through the formation of vortices. In an extreme Type‐II superconductor these vortices reduce to line singularities. Because the number of vortices is large it seems feasible to model their evolution by an averaged problem, known as the mean-field model of superconductivity. We assume that the evolution law of an individual vortex, which underlies the averaging process, involves the current of the generated magnetic field as well as the curvature vector. In the present paper we study a two‐dimensional reduction, assuming all vortices to be perpendicular to a given direction. Since both the magnetic field H and the averaged vorticity ω are curl‐free, we may represent them via a scalar magnetic potential q and a scalar stream function ψ, respectively. We study existence, uniqueness and asymptotic behaviour of solutions (ψ, q) of the resulting degenerate elliptic‐parabolic system (with curvature taken into account or not) by means of viscosity and weak solutions. In addition we relate (ψ, q) to solutions (ω, H) of the mean‐field equations without curvature. Finally we construct special solutions of the corresponding stationary equations with two or more superconducting phases. (Accepted August 8, 1997)     

Quasi‐One‐Dimensional Riemann Problems and Their Role in Self‐Similar Two‐Dimensional Problems

We study two‐dimensional Riemann problems with piecewise constant data. We identify a class of two‐dimensional systems, including many standard equations of compressible flow, which are simplified by a transformation to similarity variables. For equations in this class, a two‐dimensional Riemann problem with sectorially constant data becomes a boundary‐value problem in the finite plane. For data leading to shock interactions, this problem separates into two parts: a quasi‐one‐dimensional problem in supersonic regions, and an equation of mixed type in subsonic regions. We prove a theorem on local existence of solutions of quasi‐one‐dimensional Riemann problems. For 2 × 2 systems, we generalize a theorem of Courant & Friedrichs, that any hyperbolic state adjacent to a constant state must be a simple wave. In the subsonic regions, where the governing equation is of mixed hyperbolic‐elliptic type, we show that the elliptic part is degenerate at the boundary, with a nonlinear variant of a degeneracy first described by Keldysh. (Accepted December 4, 1997)     

On the Oppenheimer‐Volkoff Equations in General Relativity

We introduce a new formulation of the Oppenheimer‐Volkoff (O‐V) equations, a system of ordinary differential equations that models the interior of a star in general relativity, and we use this to give a completely rigorous mathematical analysis of solutions. In particular, we prove that, under mild assumptions on the equation of state, black holes never form in solutions of the O‐V equations. As a corollary, this implies that the portion of the empty‐space Schwarzschild solution inside the Schwarzschild radius cannot be obtained as a limit of O‐V solutions having non‐zero density. We also prove that if the density ρ at radius r is ever larger than where M(r) is the total mass inside radius r, then M must become negative for some positive radius. We interpret M<0 as a condition for instability because we show that if the pressure is a decreasing function of r, then M(r)<0 at some r>0 implies that the pressure tends to infinity before r=0. (Accepted October 28, 1996)     

Application of Uniform Asymptotics to the Second Painlevé Transcendent

In this work we propose a new method for investigating connection problems for the class of nonlinear second‐order differential equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort. Our method relies on finding uniform approximations of differ ential equations of the generic form as the complex‐valued parameter . The details of the treatment rely heavily on the locations of the zeros of the function F in this limit. If they are isolated, then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of F coalesce as , then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the “classical” connection problem associated with the second Painlevé transcendent. Future papers will show how the technique can be applied with very little change to the other Painlevé equations, and to the wider problem of the asymptotic behavio ur of the general solution to any of these equations. (Accepted May 15, 1997)     

Nonlinear Stability of Homogeneous Models in Newtonian Cosmology

We consider the Vlasov‐Poisson system in a cosmological setting as studied in [18] and prove nonlinear stability of homogeneous solutions against small, spatially periodic perturbations in the L ^∞‐norm of the spatial mass density. This result is connected with the question of how large scale structures such as galaxies have evolved out of the homogeneous state of the early universe. (Accepted June 28, 1996)     

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics. (Accepted July 2, 1998)     

A New Approach to Front Propagation Problems: Theory and Applications

In this paper we present a new definition for the global in time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion‐avoidance)‐type property, then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level‐set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction‐diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction‐diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations and (iii) stochastic Ising models with long-range anisotropic interactions and general spin flip dynamics. (Accepted July 8, 1996)     

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.     

A Variational Approach to Obstacle Problems for Shearable Nonlinearly Elastic Rods

We use variational methods to study obstacle problems for geometrically exact (Cosserat) theories for the planar deformation of nonlinearly elastic rods. These rods can suffer flexure, extension, and shear. There is a marked difference between the behavior of a shearable and an unshearable rod. The set of admissible deformations is not convex, because of the exact geometry used. We first investigate the fundamental question of describing contact forces, which we necessarily treat as vector‐valued Borel measures. Moreover, we introduce techniques for describing point obstacles. Then we prove existence for a very large class of problems. Finally, using nonsmooth analysis for handling the obstacle, we show that the Euler‐Lagrange equations are satisfied almost everywhere. These equations provide very detailed structural information about the contact forces. Accepted June 3, 1996     

Stable Nucleation for the Ginzburg-Landau System with an Applied Magnetic Field

We consider the Ginzburg‐Landau system with an applied magnetic field and analyze the behavior of solutions when the domain is a cylinder (of radius ) and the applied field is parallel to the axis. It is shown that there is an upper critical value such that if the modulus of the applied field is greater than , the normal (nonsuperconducting) state (in which the order parameter is identically zero) is stable and if the modulus of the applied field is slightly below , the normal state is unstable. In addition, it is shown that there is a positive lower critical value such that the normal state is unstable if the modulus of the applied field is less than and stable if the modulus is slightly above . In the case of type‐II materials for whic h the Ginzburg‐Landau constant κ is large, it is shown that there is a discrete set of radii ℬ(κ) such that if and is sufficiently large, then for each applied field of modulus slightly less than (or slightly more than ) there is precisely one small superconducting solution (up to a gauge transformation) which is stable. Moreover for this solution, the complex‐valued order parameter ψ is zero only on the axis of the cylinder, and its winding number is proportional to the product of κ² and the cross‐sectional area of the cylinder. In addition, the solution exhibits “surface superconductivity” as predicted by the physicists de Gennes and St. James. (Accepted July 15, 1996)     

Symmetry of Ground States of Quasilinear Elliptic Equations

. We consider the problem of radial symmetry for non‐negative continuously differentiable weak solutions of elliptic equations of the form under the ground state condition Using the well‐known moving plane method of Alexandrov and Serrin, we show, under suitable conditions on A and f, that all ground states of (1) are radially symmetric about some origin O in . In particular, we obtain radial symmetry for compactly supported ground states and give sufficient conditions for the positivity of ground states in terms of the given operator A and the nonlinearity f. (Accepted September 21, 1998)