Annales De L Institut Henri Poincare-Analyse Non Lineaire最新文献

{"title":"Long time confinement of vorticity around a stable stationary point vortex in a bounded planar domain","authors":"Martin Donati,&nbsp;Dragoș Iftimie","doi":"10.1016/j.anihpc.2020.11.009","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.009","url":null,"abstract":"<div><p><span>In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in </span><span>[2]</span><span> remains true in the case of a stationary point vortex in a simply-connected bounded domain<span>. The domain and the stationary point vortex must satisfy a condition expressed in terms of the conformal mapping from the domain to the unit disk. Explicit examples are discussed at the end.</span></span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.11.009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form","authors":"Baptiste Trey","doi":"10.1016/j.anihpc.2020.11.002","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.002","url":null,"abstract":"<div><p>In this paper we consider minimizers of the functional<span><span><span><math><mi>min</mi><mo>⁡</mo><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>+</mo><mi>Λ</mi><mo>|</mo><mi>Ω</mi><mo>|</mo><mo>,</mo><mspace></mspace><mo>:</mo><mspace></mspace><mi>Ω</mi><mo>⊂</mo><mi>D</mi><mtext> open</mtext><mo>}</mo></math></span></span></span> where <span><math><mi>D</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is a bounded open set and where <span><math><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> are the first <em>k</em><span><span> eigenvalues on Ω of an operator in divergence form with </span>Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets </span><span><math><msup><mrow><mi>Ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> have finite perimeter and that their free boundary <span><math><mo>∂</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∩</mo><mi>D</mi></math></span> is composed of a <em>regular part</em>, which is locally the graph of a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-regular function, and a <span><em>singular part</em></span>, which is empty if <span><math><mi>d</mi><mo>&lt;</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, discrete if <span><math><mi>d</mi><mo>=</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span><span> and of Hausdorff dimension at most </span><span><math><mi>d</mi><mo>−</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> if <span><math><mi>d</mi><mo>&gt;</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, for some <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><mo>{</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>}</mo></math></span>.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.11.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence rate for the incompressible limit of nonlinear diffusion–advection equations","authors":"Noemi David, Tomasz Dkebiec, B. Perthame","doi":"10.4171/aihpc/53","DOIUrl":"https://doi.org/10.4171/aihpc/53","url":null,"abstract":"The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cell-population models to free boundary problems of Hele-Shaw type. Although vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV -uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces. 2010 Mathematics Subject Classification. 35K57; 35K65; 35Q92; 35B45;","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74365479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantitative homogenization for combustion in random media","authors":"Y. Zhang, Andrej Zlatoš","doi":"10.4171/aihpc/80","DOIUrl":"https://doi.org/10.4171/aihpc/80","url":null,"abstract":"We obtain the first quantitative stochastic homogenization result for reaction-diffusion equations, for ignition reactions in dimensions $dle 3$ that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton-Jacobi equations.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76789019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Singularities in $L^1$-supercritical Fokker–Planck equations: A qualitative analysis","authors":"Katharina Hopf","doi":"10.4171/aihpc/85","DOIUrl":"https://doi.org/10.4171/aihpc/85","url":null,"abstract":"A class of nonlinear Fokker-Planck equations with superlinear drift is investigated in the $L^1$-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis with main focus on the isotropic case, where solutions will be shown to converge to the unique minimiser of the free energy as time tends to infinity. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90417288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modified energies for the periodic generalized KdV equation and applications","authors":"F. Planchon, N. Tzvetkov, N. Visciglia","doi":"10.4171/aihpc/62","DOIUrl":"https://doi.org/10.4171/aihpc/62","url":null,"abstract":"We construct modified energies for the generalized KdV equation. As a consequence, we obtain quasi-invariance of the high order Gaussian measures along with L regularity on the corresponding RadonNykodim density, as well as new bounds on the growth of the Sobolev norms of the solutions.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84861263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gevrey regularity for the Vlasov-Poisson system","authors":"Renato Velozo Ruiz","doi":"10.1016/j.anihpc.2020.10.006","DOIUrl":"10.1016/j.anihpc.2020.10.006","url":null,"abstract":"<div><p>We prove propagation of <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey regularity <span><math><mo>(</mo><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>)</mo></math></span> for the Vlasov-Poisson system on <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> using a Fourier space method in analogy to the results proved for the 2D-Euler system in <span>[20]</span> and <span>[23]</span>. More precisely, we give quantitative estimates for the growth of the <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey norm and decay of the regularity radius for the solution of the system in terms of the force field and the volume of the support in the velocity variable of the distribution of matter. As an application, we show global existence of <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey solutions (<span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>) for the Vlasov-Poisson system in <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Furthermore, the propagation of Gevrey regularity can be easily modified to obtain the same result in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In particular, this implies global existence of analytic <span><math><mo>(</mo><mi>s</mi><mo>=</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey solutions (<span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>) for the Vlasov-Poisson system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84409593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system","authors":"Frédéric Rousset,&nbsp;Changzhen Sun","doi":"10.1016/j.anihpc.2020.11.004","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.004","url":null,"abstract":"<div><p><span>We prove a stability result of constant equilibria<span> for the three dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter </span></span><em>ε</em> while the incompressible part of the initial velocity is assumed to be small compared to <em>ε</em>. We then get a unique global smooth solution. We also prove a uniform in <em>ε</em> time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at <em>ε</em> fixed and the dispersive techniques (dispersive estimates and normal forms) that are useful for the inviscid irrotational system.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.11.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91601825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers","authors":"Dongyuan Xiao,&nbsp;Ryunosuke Mori","doi":"10.1016/j.anihpc.2020.09.007","DOIUrl":"10.1016/j.anihpc.2020.09.007","url":null,"abstract":"<div><p>In this paper, we investigate the spreading properties of solutions of farmer and hunter-gatherer model which is a three-component reaction-diffusion system. Ecologically, the model describes the geographical spreading of an initially localized population of farmers into a region occupied by hunter-gatherers. This model was proposed by Aoki, Shida and Shigesada in 1996. By numerical simulations and some formal linearization arguments, they concluded that there are four different types of spreading behaviors depending on the parameter values. Despite such intriguing observations, no mathematically rigorous studies have been made to justify their claims. The main difficulty comes from the fact that the comparison principle does not hold for the entire system. In this paper, we give theoretical justification to all of the four types of spreading behaviors observed by Aoki et al. Furthermore, we show that a logarithmic phase drift of the front position occurs as in the scalar KPP equation. We also investigate the case where the motility of the hunter-gatherers is larger than that of the farmers, which is not discussed in the paper of Aoki et al.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83267247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities","authors":"Tianling Jin ,&nbsp;Jingang Xiong","doi":"10.1016/j.anihpc.2020.10.005","DOIUrl":"10.1016/j.anihpc.2020.10.005","url":null,"abstract":"<div><p>We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ⁎ inf type Harnack inequality of Schoen for integral equations.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89291508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}