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

{"title":"Local limit of nonlocal traffic models: Convergence results and total variation blow-up","authors":"Maria Colombo ,&nbsp;Gianluca Crippa ,&nbsp;Elio Marconi ,&nbsp;Laura V. Spinolo","doi":"10.1016/j.anihpc.2020.12.002","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.12.002","url":null,"abstract":"<div><p><span><span>Consider a nonlocal conservation law where the flux function depends on the </span>convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the </span>initial datum<span> satisfies a one-sided Lipschitz condition and is bounded away from 0. We also exhibit a counter-example showing that, if the initial datum attains the value 0, then there are severe obstructions to a convergence proof.</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.2020.12.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291873","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 the controllability and stabilization of the Benjamin equation on a periodic domain","authors":"M. Panthee,&nbsp;F. Vielma Leal","doi":"10.1016/j.anihpc.2020.12.004","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.12.004","url":null,"abstract":"<div><p>The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain <span><math><mi>T</mi></math></span>. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><mi>T</mi><mo>)</mo></math></span>, with <span><math><mi>s</mi><mo>≥</mo><mn>0</mn></math></span>. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in <span>[32]</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.2020.12.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291875","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":"Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity","authors":"Xu Yuan","doi":"10.1016/j.anihpc.2020.11.008","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.008","url":null,"abstract":"<div><p>We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity<span><span><span><math><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>−</mo><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>+</mo><mi>u</mi><mo>−</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo>+</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mtext>on</mtext><mspace></mspace><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>×</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mn>1</mn><mo>&lt;</mo><mi>q</mi><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mo>∞</mo></math></span>. The main result states the stability in the energy space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>×</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span><span> of the sums of decoupled solitary waves with different speeds, up to the natural instabilities. The proof is inspired by the techniques developed for the generalized Korteweg-de Vries equation and the nonlinear Schrödinger equation in a similar context by Martel, Merle and Tsai </span><span>[14]</span>, <span>[15]</span>. However, the adaptation of this strategy to a wave-type equation requires the introduction of a new energy functional adapted to the Lorentz transform.</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.2020.11.008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291878","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":"Edge-localized states on quantum graphs in the limit of large mass","authors":"Gregory Berkolaiko ,&nbsp;Jeremy L. Marzuola ,&nbsp;Dmitry E. Pelinovsky","doi":"10.1016/j.anihpc.2020.11.003","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.003","url":null,"abstract":"<div><p><span><span>We construct and quantify asymptotically in the limit of large mass a variety of edge-localized stationary states of the focusing nonlinear Schrödinger equation on a quantum graph. The method is applicable to general bounded and unbounded graphs. The solutions are constructed by matching a localized large amplitude </span>elliptic function on a single edge with an exponentially smaller remainder on the rest of the graph. This is done by studying the intersections of Dirichlet-to-Neumann manifolds (nonlinear analogues of Dirichlet-to-Neumann maps) corresponding to the two parts of the graph. For the quantum graph with a given set of pendant, looping, and internal edges, we find the edge on which the state of smallest energy at fixed mass is localized. </span>Numerical studies of several examples are used to illustrate the analytical results.</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.2020.11.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291876","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":"Phase transitions on the Markov and Lagrange dynamical spectra","authors":"Davi Lima ,&nbsp;Carlos Gustavo Moreira","doi":"10.1016/j.anihpc.2020.11.007","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.007","url":null,"abstract":"<div><p>The Markov and Lagrange dynamical spectra were introduced by Moreira and share several geometric and topological aspects with the classical ones. However, some features of generic dynamical spectra associated to hyperbolic sets can be proved in the dynamical case and we do not know if they are true in classical case.</p><p>They can be a good source of natural conjectures about the classical spectra: it is natural to conjecture that some properties which hold for generic dynamical spectra associated to hyperbolic maps also hold for the classical Markov and Lagrange spectra.</p><p>In this paper, we show that, for generic dynamical spectra associated to horseshoes, there are transition points <em>a</em> and <span><math><mover><mrow><mi>a</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> in the Markov and Lagrange spectra respectively, such that for any <span><math><mi>δ</mi><mo>&gt;</mo><mn>0</mn></math></span>, the intersection of the Markov spectrum with <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>a</mi><mo>−</mo><mi>δ</mi><mo>)</mo></math></span><span> has Hausdorff dimension smaller than one, while the intersection of the Markov spectrum with </span><span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>+</mo><mi>δ</mi><mo>)</mo></math></span> has non-empty interior. Similarly, the intersection of the Lagrange spectrum with <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>−</mo><mi>δ</mi><mo>)</mo></math></span> has Hausdorff dimension smaller than one, while the intersection of the Lagrange spectrum with <span><math><mo>(</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>,</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>+</mo><mi>δ</mi><mo>)</mo></math></span> has non-empty interior. We give an open set of examples where <span><math><mi>a</mi><mo>≠</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> and we prove that, in the conservative case, generically, <span><math><mi>a</mi><mo>=</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> and, for any <span><math><mi>δ</mi><mo>&gt;</mo><mn>0</mn></math></span>, the intersection of the Lagrange spectrum with <span><math><mo>(</mo><mi>a</mi><mo>−</mo><mi>δ</mi><mo>,</mo><mi>a</mi><mo>)</mo></math></span> has Hausdorff dimension one.</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.2020.11.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291881","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":"Multiphase free discontinuity problems: Monotonicity formula and regularity results","authors":"Dorin Bucur ,&nbsp;Ilaria Fragalà ,&nbsp;Alessandro Giacomini","doi":"10.1016/j.anihpc.2020.12.003","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.12.003","url":null,"abstract":"<div><p>The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.</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.2020.12.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291871","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":"Critical chirality in elliptic systems","authors":"Francesca Da Lio,&nbsp;Tristan Rivière","doi":"10.1016/j.anihpc.2020.11.006","DOIUrl":"10.1016/j.anihpc.2020.11.006","url":null,"abstract":"<div><p>We establish the regularity in 2 dimension of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span> solutions to critical elliptic systems in divergence form involving chirality operators of finite </span><span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup></math></span>-energy.</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.2020.11.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81655585","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":"Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons","authors":"Xavier Friederich","doi":"10.1016/j.anihpc.2020.11.010","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.010","url":null,"abstract":"<div><p>We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non dispersion.</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.2020.11.010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72264041","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 critical points of the relative fractional perimeter","authors":"Andrea Malchiodi ,&nbsp;Matteo Novaga ,&nbsp;Dayana Pagliardini","doi":"10.1016/j.anihpc.2020.11.005","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.11.005","url":null,"abstract":"<div><p>We study the localization of sets with constant nonlocal mean curvature<span> and prescribed small volume in a bounded open set, proving that they are sufficiently close to critical points of a suitable nonlocal potential. We then consider the fractional perimeter in half-spaces. We prove existence of minimizers under fixed volume constraint, and we show some properties such as smoothness and rotational symmetry.</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.2020.11.005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72264104","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 unique continuation principles for some elliptic systems","authors":"Ederson Moreira dos Santos ,&nbsp;Gabrielle Nornberg ,&nbsp;Nicola Soave","doi":"10.1016/j.anihpc.2020.12.001","DOIUrl":"https://doi.org/10.1016/j.anihpc.2020.12.001","url":null,"abstract":"<div><p><span>In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully </span>nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.12.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72291874","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}