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Aggregationdiffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191106
Katy Craig; Ihsan TopalogluInspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of wellknown aggregationdiffusion energies. We show that minimizers of aggregationdiffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of

Global classical solutions to quadratic systems with mass control in arbitrary dimensions Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191004
Klemens Fellner; Jeff Morgan; Bao Quoc TangThe global existence of classical solutions to reactiondiffusion systems in arbitrary space dimensions is studied. The nonlinearities are assumed to be quasipositive, to have (slightly super) quadratic growth, and to possess a mass control, which includes the important cases of mass conservation and mass dissipation. Under these assumptions, the local classical solution is shown to be global, and

A higher speed type II blowup for the five dimensional energy critical heat equation Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191126
Junichi HaradaThis paper is concerned with blowup solutions of the five dimensional energy critical heat equation ut=Δu+u43u. A goal of this paper is to show the existence of type II blowup solutions which behave as ‖u(t)‖∞∼(T−t)−3k (k=2,3,⋯). These solutions are the same ones formally derived by Filippas, Herrero and Velázquez [8]. We find a mistake in their blowup rate and correct it.

On the existence of dual solutions for Lorentzian cost functions Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191017
Martin Kell; Stefan SuhrThe dual problem of optimal transportation in LorentzFinsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of

Wellposedness issues on the periodic modified Kawahara equation Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191004
Chulkwang KwakThis paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on T), which is wellknown as a model of capillarygravity waves in an infinitely long canal over a flat bottom in a long wave regime [26]. We show in this paper some wellposedness results, mainly the global wellposedness in L2(T). The proof basically relies on the idea introduced in TakaokaTsutsumi's works

The energycritical nonlinear wave equation with an inversesquare potential Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191008
Changxing Miao; Jason Murphy; Jiqiang ZhengWe study the energycritical nonlinear wave equation in the presence of an inversesquare potential in dimensions three and four. In the defocussing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.

New concentration phenomena for a class of radial fully nonlinear equations Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200320
Giulio Galise; Alessandro Iacopetti; Fabiana Leoni; Filomena PacellaWe study radial signchanging solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values

Convergence to equilibrium for the solution of the full compressible NavierStokes equations Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20190925
Zhifei Zhang; Ruizhao ZiWe study the convergence to equilibrium for the full compressible NavierStokes equations on the torus T3. Under the conditions that both the density ρ and the temperature θ possess uniform in time positive lower and upper bounds, it is shown that global regular solutions converge to equilibrium with exponential rate. We improve the previous result obtained by Villani in (2009) [28] on two levels:

Fractional Piola identity and polyconvexity in fractional spaces Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200319
José C. Bellido; Javier Cueto; Carlos MoraCorralIn this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption of polyconvexity of the energy density, and, as a consequence, the existence of solutions to the associated Euler–Lagrange system of nonlinear fractional PDE. The

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200319
José A. Carrillo; YoungPil ChoiWe study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the socalled aggregation equation, by employing 2Wasserstein distance. By introducing an intermediate system, given by the

Analyticity and large time behavior for the Burgers equation and the quasigeostrophic equation, the both with the critical dissipation Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200319
Tsukasa IwabuchiThis paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The wellposedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as

How a minimal surface leaves a thin obstacle Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200319
Matteo Focardi; Emanuele SpadaroWe prove the optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.

Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200318
B. Lods; M. MokhtarKharroubi; R. RudnickiThis paper deals with collisionless transport equations in bounded open domains Ω⊂Rd (d⩾2) with C1 boundary ∂Ω, orthogonally invariant velocity measure m(dv) with support V⊂Rd and stochastic partly diffuse boundary operators H relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C0semigroups (UH(t))t⩾0 on L1(Ω×V,dx⊗m(dv)). We give a general

Exact controllability of semilinear heat equations in spaces of analytic functions Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200318
Camille Laurent; Lionel RosierIt is by now well known that the use of Carleman estimates allows to establish the controllability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. In this paper, we pursue the study of the reachable states of parabolic equations based on a direct approach using control inputs in Gevrey spaces by

Convex integration solutions to the transport equation with full dimensional concentration Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200318
Stefano Modena; Gabriel SattigWe construct infinitely many incompressible Sobolev vector fields u∈CtWx1,p˜ on the periodic domain Td for which uniqueness of solutions to the transport equation fails in the class of densities ρ∈CtLxp, provided 1/p+1/p˜>1+1/d. The same result applies to the transportdiffusion equation, if, in addition, p′

Thin film liquid crystals with oblique anchoring and boojums Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200318
Stan Alama; Lia Bronsard; Dmitry GolovatyWe study a twodimensional variational problem which arises as a thinfilm limit of the Landaude Gennes energy of nematic liquid crystals. We impose an oblique angle condition for the nematic director on the boundary, via boundary penalization (weak anchoring.) We show that for strong anchoring strength (relative to the usual GinzburgLandau length scale parameter,) defects will occur in the interior

Unique continuation principles in cones under nonzero Neumann boundary conditions Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200204
Serena Dipierro; Veronica Felli; Enrico ValdinociWe consider an elliptic equation in a cone, endowed with (possibly inhomogeneous) Neumann conditions. The operator and the forcing terms can also allow nonLipschitz singularities at the vertex of the cone. In this setting, we provide unique continuation results, both in terms of interior and boundary points. The proof relies on a suitable Almgrentype frequency formula with remainders. As a byproduct

Lipschitz regularity for viscous HamiltonJacobi equations with Lp terms Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200204
Marco Cirant; Alessandro GoffiWe provide Lipschitz regularity for solutions to viscous timedependent HamiltonJacobi equations with righthand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (FokkerPlanck) equation. Here, the regularizing effect is due to the nondegenerate diffusion and coercivity of the Hamiltonian

Nondivergence form quasilinear heat equations driven by spacetime white noise Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200121
Máté GerencsérWe give a WongZakai type characterisation of the solutions of quasilinear heat equations driven by spacetime white noise in 1+1 dimensions. In order to show that the renormalisation counterterms are local in the solution, a careful arrangement of a few hundred terms is required. The main tool in this computation is a general ‘integration by parts’ formula that provides a number of linear identities

A counterexample to the Liouville property of some nonlocal problems Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200116
Julien Brasseur; Jérôme CovilleIn this paper, we construct a counterexample to the Liouville property of some nonlocal reactiondiffusion equations of the form∫RN∖KJ(x−y)(u(y)−u(x))dy+f(u(x))=0,x∈RN∖K, where K⊂RN is a bounded compact set, called an “obstacle”, and f is a bistable nonlinearity. When K is convex, it is known that solutions ranging in [0,1] and satisfying u(x)→1 as x→∞ must be identically 1 in the whole space. We

Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200116
Luigi D'Onofrio; Luigi Greco; KarlMikael Perfekt; Carlo Sbordone; Roberta SchiattarellaGiven a Banach space E with a supremumtype norm induced by a collection of operators, we prove that E is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space B introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual B⁎, the biduality

On the bilinear control of the GrossPitaevskii equation Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200115
Thomas Chambrion; Laurent ThomannIn this paper we study the bilinearcontrol problem for the linear and nonlinear Schrödinger equation with harmonic potential. By the means of different examples, we show how spacetime smoothing effects (Strichartz estimates, Kato smoothing effect) enjoyed by the linear flow, can help to prove obstructions to controllability.

Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200115
ClaudeMichel Brauner; Luca Lorenzi; Mingmin ZhangIn this paper we analyze the stability of the traveling wave solution for an ignitiontemperature firstorder reaction model of diffusionalthermal combustion in the case of high Lewis numbers (Le>1). In contrast to conventional Arrhenius kinetics where the reaction zone is infinitely thin, the reaction zone for stepwise temperature kinetics is of order unity. The system of two parabolic PDEs is characterized

Nonlinear instability in Vlasov type equations around rough velocity profiles Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20200109
Aymeric BaradatIn the VlasovPoisson equation, every configuration which is homogeneous in space provides a stationary solution. Penrose gave in 1960 a criterion for such a configuration to be linearly unstable. While this criterion makes sense in a measurevalued setting, the existing results concerning nonlinear instability always suppose some regularity with respect to the velocity variable. Here, thanks to a

Wellposedness of semilinear heat equations in L1 Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191230
R. Laister; M. SierżęgaThe problem of obtaining necessary and sufficient conditions for local existence of nonnegative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term f has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in L1, a necessary and sufficient integral condition on f emerged. Here, subject to this integral

Dissipative measurevalued solutions for general conservation laws Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191126
Piotr Gwiazda; Ondřej Kreml; Agnieszka ŚwierczewskaGwiazdaIn the last years measurevalued solutions started to be considered as a relevant notion of solutions if they satisfy the socalled measurevalued – strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and

Degenerate nonlocal CahnHilliard equations: Wellposedness, regularity and local asymptotics Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191106
Elisa Davoli; Helene Ranetbauer; Luca Scarpa; Lara TrussardiExistence and uniqueness of solutions for nonlocal CahnHilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal CahnHilliard

Symbolic dynamics for one dimensional maps with nonuniform expansion Ann. I. H. Poincaré – AN (IF 2.201) Pub Date : 20191106
Yuri LimaGiven a piecewise C1+β map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, for each χ>0 we construct a finitetoone Hölder continuous map from a countable topological Markov shift