Fast Diffusion to Self-Similarity: Complete Spectrum,Long-Time Asymptotics,and Numerology

The complete spectrum is determined for the operator on the Sobolev space W^1,2(Rⁿ) formed by closing the smooth functions of compact support with respect to the norm Here the Barenblatt profile is the stationary attractor of the rescaled diffusion equation in the fast, supercritical regime m the same diffusion dynamics represent the steepest descent down an entropy E(u) on probability measures with respect to the Wasserstein distance d₂. Formally, the operator H=HessE is the Hessian of this entropy at its minimum , so the spectral gap H:=2–n(1–m) found below suggests the sharp rate of asymptotic convergence: from any centered initial data 0u(0,x)L¹(Rⁿ) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t,x)=R(t)^–n(x/R(t)) is always slowest to converge. The higher eigenfunctions – which are polynomials with hypergeometric radial parts – and the presence of continuous spectrum yield additional insight into the relations between symmetries of Rⁿ and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of .Dedicated to Elliott H. Lieb on the occasion of his 70th birthday.     

Well-Posedness for Hyperbolic Systems of Conservation Laws with Large BV Data

We study the Cauchy problem for a strictly hyperbolic n×n system of conservation laws in one space dimension assuming that the initial data has bounded but possibly large total variation. Under a linearized stability condition on the Riemann problems generated by the jumps in we prove existence and uniqueness of a (local in time) BV solution, depending continuously on the initial data in L¹_loc. The last section contains an application to the 3×3 system of gas dynamics.     

Maxwell’s Equations with Vector Hysteresis

Electromagnetic processes in magnetic materials are described by Maxwells equations. In ferrimagnetic insulators, assuming that D = E, we have the equationIn ferromagnetic metals, neglecting displacement currents and assuming Ohms law, we instead getAlternatively, under quasi-stationary conditions, for either material we can also deal with the magnetostatic equations:(Here f_ext and J_ext are prescribed time-dependent fields.) In any of these settings, the dependence of M on H is represented by a constitutive law accounting for hysteresis: M= (H), being a vector extension of the relay model. This is characterized by a rectangular hysteresis loop in a prescribed x-dependent direction, and accounts for high anisotropy and nonhomogeneity. The discontinuity in this constitutive relation corresponds to the possible occurrence of free boundaries.Weak formulations are provided for Cauchy problems associated with the above equations; existence of a solution is proved via approximation by time-discretization, derivation of energy-type estimates, and passage to the limit. An analogous representation is given for hysteresis in the dependence of P on E in ferroelectric materials. A model accounting for coupled ferrimagnetic and ferroelectric hysteresis is considered, too.Acknowledgement This research was partly supported by the project Free boundary problems in applied sciences of Italian M.I.U.R.. I gratefully acknowledge the useful suggestions from the reviewers.     

Local Minimizers and Quasiconvexity – the Impact of Topology

The aim of this paper is to discuss the question of existence and multiplicity of strong local minimizers for a relatively large class of functionals : from a purely topological point of view. The basic assumptions on are sequential lower semicontinuity with respect to W^1,p-weak convergence and W^1,p-weak coercivity, and the target is a multiplicity bound on the number of such minimizers in terms of convenient topological invariants of the manifolds and .In the first part of the paper, we focus on the case where is non-contractible and proceed by establishing a link between the latter problem and the question of enumeration of homotopy classes of continuous maps from various skeleta of into . As this in turn can be tackled by the so-called obstruction method, it is evident that our results in this direction are of a cohomological nature.The second part is devoted to the case where =^N and is a bounded smooth domain. In particular we consider integralswhere the above assumptions on can be verified when the integrand F is quasiconvex and pointwise p-coercive with respect to the gradient argument. We introduce and exploit the notion of a topologically non-trivial domain and under this establish the first existence and multiplicity result for strong local minimizers of that in turn settles a longstanding open problem in the multi-dimensional calculus of variations as described in [6].     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Regularity for Infinity Harmonic Functions in Two Dimensions

A continuous function is said to be infinity harmonic if it satisfies the PDEin the viscosity sense. In this paper we prove that infinity harmonic functions are continuously differentiable when n=2.     

A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications

We prove a stability result for a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a Γ-convergence result for energies of the form where S(u) is the jump set of u and is a sequence of rectifiable sets with We prove that no interaction occurs in the Γ-limit process between the bulk and the surface part of the energy. Relying on this result, we introduce a new notion of convergence for (N−1)-rectifiable sets called σ-convergence, which is useful in the study of the stability of unilateral minimality properties.     

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Let be a domain. Suppose that f ∈ W^1,1_loc(Ω,R²) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J_f. We show that f^-1 ∈ W^1,1_loc(f(Ω),R²) and that Df⁻¹(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f⁻¹ ∈ W^1,q(f(Ω),R²) for some 1<q≤2 are also given.     

Singular Limits of a Two-Dimensional Boundary Value Problem Arising in Corrosion Modelling

We consider the boundary value problem where Ω is a smooth and bounded domain in ℝ² and λ > 0. We prove that for any integer k ≧ 1 there exist at least two solutions u _λ with the property that the boundary flux satisfies up to subsequences λ → 0, where the ξ _j are points of ∂Ω ordered clockwise in j.     

The Dirichlet Problem for H-Systems with Small Boundary Data: BlowUp Phenomena and Nonexistence Results

Given H:ℝ³→ℝ of class C¹ and bounded, we consider a sequence (uⁿ) of solutions of the H-system in the unit open disc satisfying the boundary condition uⁿ=γⁿ on ∂. In the first part of this paper, assuming that (uⁿ) is bounded in H¹(,ℝ³) we study the behavior of (uⁿ) when the boundary data γⁿ shrink to zero. We show that either uⁿ→0 strongly in H¹(,ℝ³) or uⁿ blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on ℝ². Under additional assumptions on H, we can obtain more precise information on the blow up. In the second part of this paper we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum. We detect a family of nonconstant functions H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a ``large' solution at a mountain pass level when the boundary datum is small.     

A Sharp-Interface Limit for a Two-Well Problem in Geometrically Linear Elasticity

In the theory of solid-solid phase transitions the deformation of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study Frame indifference, within a linearized setting, requires that W depends only on the symmetric part of ∇u. The potential W is nonnegative and vanishes on two wells, i.e., for d = 2, on two lines in the space of matrices. We determine, for d = 2, the Gamma limit I₀ = Γ− lim _ɛ→0 I_ɛ. The limit I₀[u] is finite only for deformations u that fulfill W(∇u)=0 almost everywhere and have sharp interfaces where ∇u has jumps. For these u, I₀[u] equals the line integral over the interfaces of a surface energy density.     

Determining Modes, Nodes and Volume Elements for Stationary Solutions of the Navier-Stokes Problem Past a Three-Dimensional Body

In this paper we show that every solution of the three-dimensional exterior Navier-Stokes boundary-value problem, corresponding to a given non-zero, constant velocity at infinity (flow past a body) and belonging to a very general functional class, , can be determined by a finite number of parameters. Our results extend the analogous classical results by Foiaş & Temam [6, 7], and by Jones & Titi [14] for the interior problem. This extension is by no means trivial, in that all fundamental tools used in the case of the interior problem – such as compactness of the Sobolev embeddings, Poincaré's inequality, and the special basis constituted by eigenfunctions of the Stokes operator – are no longer available for the exterior problem. An important consequence of our results is that any solution in is uniquely determined by the knowledge of the associated velocity field only ``near' the boundary. Just how ``near' it has to be depends only on the Reynolds number and on the body. Dedicated to John Heywood on the occasion of his 65th birthday     

Lyapunov Functional Method for 1D Radiative and Reactive Viscous Gas Dynamics

We consider the compressible Navier–Stokes system for 1D-flows of a viscous heat-conducting gas, with the pressure law and a one-order kinetics to include radiative effects and reactive processes. The mass force and the ignition phenomenon are also taken into account. For large data and under general assumptions on the heat conductivity, we establish global-in-time bounds and exponential stabilization for solutions in L^q and H¹ norms. To this end, we construct new global Lyapunov functionals and show that they describe the dynamics of solutions for any t≧0. A short proof of the corresponding global existence is also included for completeness.     

Global Behavior of Compressible Navier-Stokes Equations with a Degenerate Viscosity Coefficient

In this paper, we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity. Precisely, the viscosity coefficient μ is proportional to ρ ^θ with , where ρ is the density, and γ > 1 is the physical constant of polytropic gas. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give the uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we estimate the stabilization rate in L ^∞ norm, (weighted) L ² norm and weighted H ¹ norm of the solution as time tends to infinity.     

Mathematical Derivation of the Continuum Limit of the Magnetic Force between Two Parts of a Rigid Crystalline Material

The topic of this paper is a mathematically rigorous derivation of the continuum limit of the magnetic force between two parts of a rigid magnetized body. For this we start from a discrete setting of magnetic dipoles fixed to a scaled Bravais lattice, The limit as l corresponds to the passage to the continuum. The magnetic dipole moments are scaled in such a way that we obtain a finite total magnetic moment per unit volume. Under certain regularity assumptions on the magnetization and the boundaries we derive a force formula in the passage from the discrete setting to the continuum. Compared with a corresponding magnetic-force formula which has been previously discussed in the literature, the limiting force consists of an additional explicit local surface term, which is due to short-range effects and which reflects the lattice approximation of the underlying hypersingular integral.     

Bifurcation From Stability to Instability for a Free Boundary Problem Arising in a Tumor Model

We consider a time-dependent free boundary problem with radially symmetric initial data: σ_t − Δσ + σ = 0 if and σ(r,0)=σ₀(r) in {r < R(0)} where R(0) is given. This is a model for tumor growth, with nutrient concentration (or tumor cells density) σ(r,t) and proliferation rate then there exists a unique stationary solution (σ_S(r), R_S), where R_S depends only on the number . We prove that there exists a number μ_*, such that if μ < μ_* . . . then the stationary solution is stable with respect to non-radially symmetric perturbations, whereas if μ > μ_* then the stationary solution is unstable.     

Crystalline Mean Curvature Flow of Convex Sets

We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in . This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φ-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat φ-curvature flow starting from a compact convex set is unique.     

The Singular Set of Minima of Integral Functionals

In this paper we provide upper bounds for the Hausdorff dimension of the singular set of minima of general variational integrals where F is suitably convex with respect to Dv and Hölder continuous with respect to (x,v). In particular, we prove that the Hausdorff dimension of the singular set is always strictly less than n, where .     

The Singular Set of <Emphasis Type="Italic">ω</Emphasis>-minima

We consider ω-minima of convex variational integrals in the vectorial case n,N≥2, and we provide estimates for the Hausdorff dimension of their singular sets.     

Towards a Unified Field Theory for Classical Electrodynamics

In this paper we introduce a model which describes the relation of matter and the electromagnetic field from a unitarian standpoint in the spirit of ideas of Born and Infeld. In this model, based on a semilinear perturbation of Maxwell equations, the particles are finite-energy solitary waves due to the presence of the nonlinearity. In this respect the matter and the electromagnetic field have the same nature. Finite energy means that particles have finite mass and this makes electrodynamics consistent with the special relativity. We analyze the invariants of the motion of the semilinear Maxwell equations (SME) and their static solutions. In the magnetostatic case (i.e., when the electric field E = 0 and the magnetic field H does not depend on time) SME are reduced to the semilinear equation where × denotes the curloperator, f is the gradient of a strictly convex smooth function f:R³R and A:R³R³ is the gauge potential related to the magnetic field H (H = × A). Due to the presence of the curl operator, (1) is a strongly degenerate elliptic equation. Moreover, physical considerations impel f to be flat at zero (f(0)=0) and this fact leads us to study the problem in a functional setting related to the Orlicz space L^p+L^q. The existence of a nontrivial finite- energy solution of (1) is proved under suitable growth conditions on f. The proof is carried out by using a suitable variational framework related to the Hodge splitting of the vector field A.We thank Marino Badiale and Charles Stuart for their useful suggestions.     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla