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

{"title":"Solutions to the non-cutoff Boltzmann equation in the grazing limit","authors":"Renjun Duan, Ling-Bing He, Y. Tong, Yu-Long Zhou","doi":"10.4171/aihpc/72","DOIUrl":"https://doi.org/10.4171/aihpc/72","url":null,"abstract":"It is known that in the parameters range $-2 leq gamma<-2s$ spectral gap does not exist for the linearized Boltzmann operator without cutoff but it does for the linearized Landau operator. This paper is devoted to the understanding of the formation of spectral gap in this range through the grazing limit. Precisely, we study the Cauchy problems of these two classical collisional kinetic equations around global Maxwellians in torus and establish the following results that are uniform in the vanishing grazing parameter $epsilon$: (i) spectral gap type estimates for the collision operators; (ii) global existence of small-amplitude solutions for initial data with low regularity; (iii) propagation of regularity in both space and velocity variables as well as velocity moments without smallness; (iv) global-in-time asymptotics of the Boltzmann solution toward the Landau solution at the rate $O(epsilon)$; (v) continuous transition of decay structure of the Boltzmann operator to the Landau operator. In particular, the result in part (v) captures the uniform-in-$epsilon$ transition of intrinsic optimal time decay structures of solutions that reveals how the spectrum of the linearized non-cutoff Boltzmann equation in the mentioned parameter range changes continuously under the grazing limit.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91265365","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":"C1,α-estimates for the near field refractor","authors":"Cristian E. Gutiérrez ,&nbsp;Federico Tournier","doi":"10.1016/j.anihpc.2020.08.002","DOIUrl":"10.1016/j.anihpc.2020.08.002","url":null,"abstract":"<div><p>We establish local <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span><span> estimates for one source near field refractors under structural assumptions on the target, and with no assumptions on the smoothness of the densities.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.08.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87831160","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":"Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition","authors":"Rui Peng ,&nbsp;Chang-Hong Wu ,&nbsp;Maolin Zhou","doi":"10.1016/j.anihpc.2020.07.006","DOIUrl":"10.1016/j.anihpc.2020.07.006","url":null,"abstract":"<div><p>This paper is concerned with the classical two-species Lotka-Volterra diffusion system with strong competition. The sharp dynamical behavior of the solution is established in two different situations: either one species is an invasive one and the other is a native one or both are invasive species. Our results seem to be the first that provide a precise spreading speed and profile for such a strong competition system. Among other things, our analysis relies on the construction of new types of supersolution<span> and subsolution, which are optimal in certain sense.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.07.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79874768","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":"The third order Benjamin-Ono equation on the torus: Well-posedness, traveling waves and stability","authors":"Louise Gassot","doi":"10.1016/j.anihpc.2020.09.004","DOIUrl":"10.1016/j.anihpc.2020.09.004","url":null,"abstract":"<div><p>We consider the third order Benjamin-Ono equation on the torus<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mrow><mo>(</mo><mo>−</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mi>u</mi><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>u</mi><mi>H</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>H</mi><mo>(</mo><mi>u</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo>)</mo><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow><mo>.</mo></math></span></span></span> We prove that for any <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>, the flow map continuously extends to <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>r</mi><mo>,</mo><mn>0</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><mi>T</mi><mo>)</mo></math></span> if <span><math><mi>s</mi><mo>≥</mo><mn>0</mn></math></span>, but does not admit a continuous extension to <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>r</mi><mo>,</mo><mn>0</mn></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msubsup><mo>(</mo><mi>T</mi><mo>)</mo></math></span> if <span><math><mn>0</mn><mo>&lt;</mo><mi>s</mi><mo>&lt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Moreover, we show that the extension is weakly sequentially continuous in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>r</mi><mo>,</mo><mn>0</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><mi>T</mi><mo>)</mo></math></span> if <span><math><mi>s</mi><mo>&gt;</mo><mn>0</mn></math></span>, but is not weakly sequentially continuous in <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>r</mi><mo>,</mo><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>T</mi><mo>)</mo></math></span><span>. We then classify the traveling wave solutions for the third order Benjamin-Ono equation in </span><span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>r</mi><mo>,</mo><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>T</mi><mo>)</mo></math></span> and study their orbital stability.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85181720","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":"Semilinear problems with right-hand sides singular at u = 0 which change sign","authors":"Juan Casado-Díaz ,&nbsp;François Murat","doi":"10.1016/j.anihpc.2020.09.001","DOIUrl":"10.1016/j.anihpc.2020.09.001","url":null,"abstract":"<div><p>The present paper is devoted to the study of the existence of a solution <em>u</em><span> for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at </span><span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span><span>. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at </span><span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span>, while no restriction on its growth at <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span> is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in <em>u</em>. We also show that if the right-hand side goes to infinity at zero faster than <span><math><mn>1</mn><mo>/</mo><mo>|</mo><mi>u</mi><mo>|</mo></math></span>, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is <span><math><mn>1</mn><mo>/</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>γ</mi></mrow></msup></math></span> with <span><math><mn>0</mn><mo>&lt;</mo><mi>γ</mi><mo>&lt;</mo><mn>1</mn></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":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83437848","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":"Nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FLRW geometry","authors":"Philippe G. LeFloch ,&nbsp;Changhua Wei","doi":"10.1016/j.anihpc.2020.09.005","DOIUrl":"10.1016/j.anihpc.2020.09.005","url":null,"abstract":"<div><p>We analyze the global nonlinear stability of FLRW (Friedmann-Lemaître-Robertson-Walker) spacetimes in the presence of an irrotational perfect fluid. We assume that the fluid is governed by the so-called (generalized) Chaplygin equation of state <span><math><mi>p</mi><mo>=</mo><mo>−</mo><mfrac><mrow><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>ρ</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></math></span> relating the pressure to the mass-energy density, in which <span><math><mi>A</mi><mo>&gt;</mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span><span><span> are constants. We express the Einstein equations in wave gauge as a system of coupled nonlinear wave equations and, after performing a </span>conformal transformation, we analyze the global behavior of solutions toward the future. Under small perturbations, the </span><span><math><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></math></span><span>-spacetime metric, the mass-energy density, and the velocity vector describing the geometry and fluid unknowns remain globally close to a reference FLRW solution. Our analysis provides also the precise asymptotic behavior of the perturbed solutions toward the future.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85976374","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":"The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation","authors":"Hongzi Cong ,&nbsp;Xiaoping Yuan","doi":"10.1016/j.anihpc.2020.09.006","DOIUrl":"10.1016/j.anihpc.2020.09.006","url":null,"abstract":"<div><p><span>In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain </span><span>[9]</span>.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91072609","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":"Solutions with peaks for a coagulation-fragmentation equation. Part II: Aggregation in peaks","authors":"Marco Bonacini ,&nbsp;Barbara Niethammer ,&nbsp;Juan J.L. Velázquez","doi":"10.1016/j.anihpc.2020.08.007","DOIUrl":"10.1016/j.anihpc.2020.08.007","url":null,"abstract":"<div><p>The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. In a companion paper we constructed a two-parameter family of stationary solutions concentrated in Dirac masses, and we carefully studied the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In this paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.08.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84281812","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":"Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrödinger Equation","authors":"Guillaume Ferriere","doi":"10.1016/j.anihpc.2020.09.002","DOIUrl":"10.1016/j.anihpc.2020.09.002","url":null,"abstract":"<div><p><span><span>We consider the logarithmic Schrödinger equation (logNLS) in the focusing regime. For this equation, Gaussian </span>initial data<span> remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In this paper, we construct </span></span><em>multi-solitons</em> (or <em>multi-Gaussons</em>) for logNLS, with estimates in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>∩</mo><mi>F</mi><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span>. We also construct solutions to logNLS behaving (in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>) like a sum of <em>N</em> Gaussian solutions with different speeds (which we call <em>multi-gaussian</em>). In both cases, the convergence (as <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span>) is faster than exponential. We also prove a rigidity result on these constructed multi-gaussians and multi-solitons, showing that they are the only ones with such a convergence.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75212406","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":"On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities","authors":"Sergio Frigeri","doi":"10.1016/j.anihpc.2020.08.005","DOIUrl":"10.1016/j.anihpc.2020.08.005","url":null,"abstract":"<div><p><span>We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids<span> having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in </span></span>two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.08.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81879179","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}