Showing papers in "Annales De L Institut Henri Poincare-analyse Non Lineaire in 2001"

Existence of solutions for compressible fluid models of Korteweg type

Abstract: The purpose of this work is to prove existence and uniqueness results of suitably smooth solutions for an isothermal model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985), which can be used as a phase transition model. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence for solutions when the pressure law may present spinodal regions. Uniqueness is also obtained. Assuming a lower and upper control of the density, we also show the existence of weak solutions in dimension 2 near equilibrium. Finally, referring to the work of Z. Xin (1998) in the non-capillary case, we describe some blow-up properties of smooth solutions with finite total mass.

Incompressible, inviscid limit of the compressible Navier-Stokes system

Abstract: We prove some asymptotic results concerning global (weak) solutions of compressible isentropic Navier–Stokes equations. More precisely, we establish the convergence towards solutions of incompressible Euler equations, as the density becomes constant, the Mach number goes to 0 and the Reynolds number goes to infinity.

Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

Abstract: Nonlocal conservation laws of the form ut + Lu +∇ · f( u)= 0, where −L is the generator of a Levy semigroup on L 1 (R n ), are encountered in continuum mechanics as model equations with anomalous diffusion. They are generalizations of the classical Burgers equation. We study the critical case when the diffusion and nonlinear terms are balanced, e.g. L ∼ (−�) α/2 ,1 <α< 2, f( s)∼ s|s| r−1 , r = 1 + (α − 1)/n. The results include decay rates of solutions and their genuinely nonlinear asymptotic behavior as time t tends to infinity, determined by self-similar source solutions.  2001 Editions scientifiques et medicales Elsevier

On the thermodynamic limit for Hartree–Fock type models

Abstract: We continue here our study [10–13] of the thermodynamic limit for various models of Quantum Chemistry, this time focusing on the Hartree–Fock type models. For the reduced Hartree–Fock models, we prove the existence of the thermodynamic limit for the energy per unit volume. We also define a periodic problem associated to the Hartree–Fock model, and prove that it is well-posed.

Enhancement of the traveling front speeds in reaction-diffusion equations with advection

Abstract: We establish rigorous lower bounds on the speed of traveling fronts and on the bulk burning rate in reaction-diffusion equation with passive advection. The non-linearity is assumed to be of either KPP or ignition type. We consider two main classes of flows. Percolating flows, which are characterized by the presence of long tubes of streamlines mixing hot and cold material, lead to strong speed-up of burning which is linear in the amplitude of the flow, U. On the other hand the cellular flows, which have closed streamlines, are shown to produce weaker increase in reaction. For such flows we get a lower bound which grows as U1/5 for a large amplitude of the flow.

Geometric approach to Goursat flags

Abstract: A Goursat flag is a chain Ds⊂Ds−1⊂⋯⊂D1⊂D0=TM of subbundles of the tangent bundle TM such that corankDi=i and Di−1 is generated by the vector fields in Di and their Lie brackets. Engel, Goursat, and Cartan studied these flags and established a normal form for them, valid at generic points of M. Recently Kumpera, Ruiz and Mormul discovered that Goursat flags can have singularities, and that the number of these grows exponentially with the corank s. Our Theorem 1 says that every corank s Goursat germ, including those yet to be discovered, can be found within the s-fold Cartan prolongation of the tangent bundle of a surface. Theorem 2 says that every Goursat singularity is structurally stable, or irremovable, under Goursat perturbations. Theorem 3 establishes the global structural stability of Goursat flags, subject to perturbations which fix a certain canonical foliation. It relies on a generalization of Gray's theorem for deformations of contact structures. Our results are based on a geometric approach, beginning with the construction of an integrable subflag to a Goursat flag, and the sandwich lemma which describes inclusions between the two flags. We show that the problem of local classification of Goursat flags reduces to the problem of counting the fixed points of the circle with respect to certain groups of projective transformations. This yields new general classification results and explains previous classification results in geometric terms. In the last appendix we obtain a corollary to Theorem 1. The problems of locally classifying the distribution which models a truck pulling s trailers and classifying arbitrary Goursat distribution germs of corank s+1 are the same.

Nonhomogeneous Cahn–Hilliard fluids

Abstract: In this paper we are interested in the study of a model of nonhomogeneous diphasic incompressible flow. More precisely we consider a coupling of a Cahn–Hilliard and an incompressible Navier–Stokes equations where the densities of the phases are different. For this general model we can only show the local existence of a unique very regular solution and the existence of weaker solutions is still an open problem. But, if we look at the behavior of the system when the densities tends to be equal (slightly nonhomogeneous case), we show the existence of a global weak solution and of a unique local strong solution (which is in fact global in 2D). Finally, an asymptotic stability result for the metastable states is shown in this slightly nonhomogeneous case.

On the subanalyticity of carnot-caratheodory distances

Abstract: is not!This paper is a new step in a rather long research line, see [1,5,6,9,10,15,17,20]. Themain tools are the nilpotent approximation, Morse-type indices of geodesics, both in thenormal and abnormal cases, and transversality techniques.We finish the introduction with some conjectures on still open questions.(1) Small balls

Lower semicontinuity of L∞ functionals

Abstract: We study the lower semicontinuity properties and existence of a minimizer of the functional F( u)= ess sup x∈� f (x,u(x),Du(x)) on W 1,∞ (� ; R m ). We introduce the notions of Morrey quasiconvexity, polyquasiconvexity, and rank-one quasiconvexity, all stemming from the notion of quasiconvexity (= convex level sets) of f in the last variable. We also formally derive the Aronsson-Euler equation for such problems.  2001 Editions scientifiques et medicales Elsevier SAS

Asymptotic behaviour of holomorphic strips

Abstract: The asymptotic behaviour of a finite energy pseudoholomorphic strip with Lagrangian boundary conditions in a symplectic manifold is determined by an eigenfunction of the linearized operator at the (transverse) intersection.

An example of non-convex minimization and an application to Newton's problem of the body of least resistance

Abstract: We study the minima of the functional ∫ Ω f(∇u) . The function f is not convex, the set Ω is a domain in R 2 and the minimum is sought over all convex functions on Ω with values in a given bounded interval. We prove that a minimum u is almost everywhere ‘on the boundary of convexity’, in the sense that there exists no open set on which u is strictly convex. In particular, wherever the Gaussian curvature is finite, it is zero. An important application of this result is the problem of the body of least resistance as formulated by Newton (where f ( p )=1/(1+| p | 2 ) and Ω is a ball), implying that the minimizer is not radially symmetric. This generalizes a result in [1].

Symmetry for exterior elliptic problems and two conjectures in potential theory

Abstract: In this paper we extend a classical result of Serrin to a class of elliptic problems Δu+f(u,|∇u|)=0 in exterior domains R N ⧹G (or Ω⧹G with Ω and G bounded). In case G is an union of a finite number of disjoint C2-domains Gi and u=ai>0, ∂u/∂n=αi⩽0 on ∂Gi, u→0 at infinity, we show that if a non-negative solution of such a problem exists, then G has only one component and it is a ball. As a consequence we establish two results in electrostatics and capillarity theory. We further obtain symmetry results for quasilinear elliptic equations in the exterior of a ball.

A priori estimates for solutions of fully nonlinear special lagrangian equations

Abstract: We derive an a priori C2,α estimate in dimension three for the equation F(D2u)=arctanλ1+arctanλ2+arctanλ3=c, where λ1,λ2,λ3 are the eigenvalues of the Hessian D2u. For −π/2

On the symmetry of blowup solutions to a mean field equation

Abstract: In this article, we consider the mean field equation Δu+ρeu∫eu−1A=0inΣ, where Σ is a flat torus and A is the area of Σ. This paper is concerned with the symmetry induced by the phenomenon of concentration. By using the method of moving planes, we prove that blowup solutions often possess certain symmetry. In this paper, we consider cases when solutions blowup at one or two points. We also consider related problems for annulu domains of R2.

Positive bound states having prescribed symmetry for a class of nonlinear elliptic equations in RN

Abstract: We consider positive solutions of −div(|x|−2a∇u)=|x|−bpup−1,u⩾0inRN, where for N⩾2: a<(N−2)/2,a

Local calibrations for minimizers of the Mumford–Shah functional with a regular discontinuity set

Abstract: Using a calibration method, we prove that, if w is a function which satisfies all Euler conditions for the Mumford–Shah functional on a two-dimensional open set Ω, and the discontinuity set Sw of w is a regular curve connecting two boundary points, then there exists a uniform neighbourhood U of Sw such that w is a minimizer of the Mumford–Shah functional on U with respect to its own boundary conditions on ∂U. We show that Euler conditions do not guarantee in general the minimality of w in the class of functions with the same boundary value of w on ∂Ω and whose extended graph is contained in a neighbourhood of the extended graph of w, and we give a sufficient condition in terms of the geometrical properties of Ω and Sw under which this kind of minimality holds.

Existence of critical points for some noncoercive functionals

Abstract: In this paper we study critical points problems for some integral functionals with principal part having degenerate coerciveness, whose model is J(v)= 1 2 ∫ Ω |∇v| 2 (b(x)+|v|) 2α − 1 m ∫ Ω |v| m , v∈H 0 1 (Ω), with 1 ∗ (1−α) . We will prove several existence and nonexistence results depending on different assumptions on both m and α .

Long-time vanishing properties of solutions of some semilinear parabolic equations

Abstract: We study the long-time behavior of solutions of semilinear parabolic equations of the following type (PE) ∂tu−∇.A(x,t,u,∇u)+f(x,u)=0 where f(x,u)≈b(x)|u|q−1u, b being a nonnegative bounded and measurable function and q a real number such that 0≤q<1. We give criteria which imply that any solution of the above equations vanishes in finite time and these criteria are associated to semi-classical limits of some Schrodinger operators. We also give a series of sufficient conditions on b(x) which imply that any supersolution with positive initial data does not to vanish identically for any positive t.

A Glimm type functional for a special Jin-Xin relaxation model

Abstract: We consider a special case of the Jin–Xin relaxation systems [∗]ut+vx=0,vt+λ2ux=(F(u)−v)/e. We assume that the integral curves of the eigenvectors ri of DF(u) are straight lines. In this setting we prove that for every initial data u,v with sufficiently small total variation the solution (ue,ve) of (∗) is well defined for all t>0, and its total variation satisfies a uniform bound, independent of t,e. Moreover, as e tends to 0+, the solutions (ue,ve) converge to a unique limit (u(t),v(t)): u(t) is the unique entropic solution of the corresponding hyperbolic system ut+F(u)x=0 and v(t,x)=F(u(t,x)) for all t>0, a.e. x∈R. The proofs rely on the introduction of a new functional for the solutions of (∗), corresponding to the Glimm interaction potential for the approaching waves of different families.

Existence of guided cylindrical TM-modes in a homogeneous self-focusing dielectric

Abstract: Keywords: self-trapped beam of light ; guided waves ; energy ; integral ; focusing dielectric material ; nonlinear ; eigenvalue problem ; self-trapped transverse magnetic ; field modes ; cylindrical optical filter ; Mountain Pass ; Theorem Reference ANA-ARTICLE-2001-005doi:10.1016/S0294-1449(00)00125-6View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

Regularity properties of free discontinuity sets

Abstract: This paper is concerned with the problem of estimating the dimension, expected to be N −2, of the singular set of a minimizer of a functional with free discontinuities in N dimensions. The best result already known, namely the ( N −1)-negligibility, is improved here.

Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities

Abstract: The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C 1 (∂Ω) boundary data u . We show here that the same result holds for u ∈C 0,μ (∂Ω) if μ> 1 2 and it cannot be extended to cover the case μ= 1 2 . The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers.

Nonlinear monotone semigroups and viscosity solutions

Abstract: In a celebrated paper motivated by applications to image analysis, L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel showed that any monotone semigroup defined on the space of bounded uniformly continuous functions, which satisfies suitable regularity and locality assumptions is in fact a semigroup associated to a fully nonlinear, possibly degenerate, second-order parabolic partial differential equation. In this paper, we extend this result by weakening the assumptions required on the semigroup to obtain such a result and also by treating the case where the semigroup is defined on a general space of continuous functions like, for example, a space of continuous functions with a prescribed growth at infinity. These extensions rely on a completely different proof using in a more central way the monotonicity of the semigroup and viscosity solutions methods. Then we study the consequences on the partial differential equation of various additional assumptions on the semigroup. Finally we briefly present the adaptation of our proof to the case of two-parameters families.

Exotic solutions of the conformal scalar curvature equation in Rn

Abstract: We construct global exotic solutions of the conformal scalar curvature equation Δu +[ n ( n −2)/4] Ku ( n +2)/( n −2) =0 in R n , with K ( x ) approaching 1 near infinity in order as close to the critical exponent as possible.

American prices embedded in European prices

Abstract: In this paper, we are interested in American option prices in the Black–Scholes model. For a large class of payoffs, we show that in the region where the European price increases with the time to maturity, this price is equal to the American price of another claim. We give examples in which we explicit the corresponding claims. The characterization of the American claims obtained in this way remains an open question.