Journal de Mathematiques Pures et Appliquees最新文献

{"title":"From KP-I lump solution to travelling waves of Gross-Pitaevskii equation","authors":"Yong Liu ,&nbsp;Zhengping Wang ,&nbsp;Juncheng Wei ,&nbsp;Wen Yang","doi":"10.1016/j.matpur.2025.103801","DOIUrl":"10.1016/j.matpur.2025.103801","url":null,"abstract":"<div><div>Let <em>q</em> be a nondegenerate lump type solution to the KP-I (Kadomtsev-Petviashvili-I) equation<span><span><span><math><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mi>q</mi><mo>−</mo><mn>2</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>q</mi><mo>−</mo><mn>3</mn><msqrt><mrow><mn>2</mn></mrow></msqrt><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><msup><mrow><mo>(</mo><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>−</mo><mn>2</mn><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>q</mi><mo>=</mo><mn>0</mn><mo>.</mo></math></span></span></span> We show the existence of travelling wave solutions with the form <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>, for the GP (Gross-Pitaevskii) equation<span><span><span><math><mi>i</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>Ψ</mi><mo>+</mo><mi>Δ</mi><mi>Ψ</mi><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mo>|</mo><mi>Ψ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>Ψ</mi><mo>=</mo><mn>0</mn><mspace></mspace><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> with travelling speed <span><math><mi>c</mi><mo>=</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>i</mi><mi>ε</mi><mi>q</mi><mo>+</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. This proves the existence of finite energy solutions in the so-called Jones-Roberts program within the transonic regime <span><math><mi>c</mi><mo>∈</mo><mo>(</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo></math></span>. The main ingredients in our proof are detailed point-wise estimates for the Green functions associated to a family of fourth order hypoelliptic operators. In view of the classification of lump type solutions of the KP-I equation, our proof also indicates that for fixed small <em>ε</em>, there should exist a sequence of travelling wave solutions to GP equation, with energy tends to infinity.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103801"},"PeriodicalIF":2.3,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265977","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":"Global axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations","authors":"Hui Chen ,&nbsp;Zijin Li ,&nbsp;Ping Zhang","doi":"10.1016/j.matpur.2025.103807","DOIUrl":"10.1016/j.matpur.2025.103807","url":null,"abstract":"<div><div>In this paper, we prove the global existence and uniqueness of axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations in a cylindrical domain with Navier boundary condition provided that the swirl component of the initial velocity is sufficiently small. The main idea of the proof is to perform energy estimates for the pair <span><math><mo>(</mo><mi>J</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></math></span>, where <figure><img></figure> and <figure><img></figure> is a corrector of <figure><img></figure>. In order to close the energy estimates, we introduced the derivative-reduction technique and new elliptic estimates of the pressure function, which are established to overcome difficulties arising from the lower-order terms in the Navier boundary condition. We also consider the global regularity of the axisymmetric solution to the Navier–Stokes equations with full viscosity subject to the total-slip Navier boundary condition. Several new inequalities are established to address the challenges posed by the weak horizontal diffusion of the swirl component.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103807"},"PeriodicalIF":2.3,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265978","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":"Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with Ld drift","authors":"Pilgyu Jung ,&nbsp;Kwan Woo","doi":"10.1016/j.matpur.2025.103805","DOIUrl":"10.1016/j.matpur.2025.103805","url":null,"abstract":"<div><div>We explore the higher integrability of Green's functions associated with the second-order elliptic equation <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>f</mi></math></span> in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term <span><math><mi>b</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> and the source term <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for some <span><math><mi>p</mi><mo>&lt;</mo><mi>d</mi></math></span>. This provides an alternative and analytic proof of a result by N.V. Krylov (<em>Ann. Probab.</em>, 2021) concerning <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (<em>Duke Math. J.</em>, 1984).</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103805"},"PeriodicalIF":2.3,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219418","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":"Derivation of Hartree theory for two-dimensional attractive Bose gases in almost Gross–Pitaevskii regime","authors":"Lukas Junge ,&nbsp;François L.A. Visconti","doi":"10.1016/j.matpur.2025.103800","DOIUrl":"10.1016/j.matpur.2025.103800","url":null,"abstract":"<div><div>We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103800"},"PeriodicalIF":2.3,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145264797","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":"Qualitative properties of the spreading speed of a population structured in space and in phenotype","authors":"Nathanaël Boutillon","doi":"10.1016/j.matpur.2025.103804","DOIUrl":"10.1016/j.matpur.2025.103804","url":null,"abstract":"<div><div>We consider a nonlocal Fisher-KPP equation that models a population structured in space and in phenotype. The population lives in a heterogeneous periodic environment: the diffusion coefficient, the mutation coefficient and the fitness of an individual may depend on its spatial position and on its phenotype.</div><div>We first prove a Freidlin-Gärtner formula for the spreading speed of the population. We then study the behaviour of the spreading speed in different scaling limits (small and large period, small and large mutation coefficient). Finally, we exhibit new phenomena arising thanks to the phenotypic dimension.</div><div>Our results are also valid when the phenotype is seen as another spatial variable along which the population does not spread.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103804"},"PeriodicalIF":2.3,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145264963","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":"Long time classical solutions of quasilinear Klein-Gordon equations with small weakly decaying initial data","authors":"Fei Hou ,&nbsp;Huicheng Yin","doi":"10.1016/j.matpur.2025.103803","DOIUrl":"10.1016/j.matpur.2025.103803","url":null,"abstract":"<div><div>It is well known that for the quasilinear Klein-Gordon equation with quadratic nonlinearity and sufficiently decaying small initial data, there exists a global smooth solution if the space dimensions <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. When the initial data are of size <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> in the Sobolev space, for the semilinear Klein-Gordon equation satisfying the null condition, the authors in the article (Delort and Fang, 2000 <span><span>[11]</span></span>) prove that the solution exists in time <span><math><mo>[</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>≥</mo><mi>C</mi><msup><mrow><mi>e</mi></mrow><mrow><mi>C</mi><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mi>μ</mi></mrow></msup></mrow></msup></math></span> (<span><math><mi>μ</mi><mo>=</mo><mn>1</mn></math></span> if <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mi>μ</mi><mo>=</mo><mn>2</mn><mo>/</mo><mn>3</mn></math></span> if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>). In the present paper, we will focus on the general quasilinear Klein-Gordon equation without the null condition and further show that the existence time of the solution can be improved to <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>=</mo><mo>+</mo><mo>∞</mo></math></span> if <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>≥</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>C</mi><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></msup></math></span> if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. In addition, for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> and any fixed number <span><math><mi>α</mi><mo>&gt;</mo><mn>0</mn></math></span>, if the weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm of the initial data with the weight <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup></math></span> is small, then the solution exists globally and scatters to a free solution. Our arguments are based on the introduction of a new good unknown, the Strichartz estimate, the weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm estimate and the resonance analysis.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103803"},"PeriodicalIF":2.3,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219207","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":"Waiting time solutions in gas dynamics","authors":"Juhi Jang ,&nbsp;Jiaqi Liu ,&nbsp;Nader Masmoudi","doi":"10.1016/j.matpur.2025.103806","DOIUrl":"10.1016/j.matpur.2025.103806","url":null,"abstract":"<div><div>In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent <span><math><mi>γ</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only Hölder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103806"},"PeriodicalIF":2.3,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145264964","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 magnetic Dirichlet to Neumann operator on the exterior of the disk – Diamagnetism, weak-magnetic field limit and flux effects","authors":"Bernard Helffer,&nbsp;François Nicoleau","doi":"10.1016/j.matpur.2025.103799","DOIUrl":"10.1016/j.matpur.2025.103799","url":null,"abstract":"<div><div>In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>(</mo><mi>b</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> on the exterior of the disk with respect to a magnetic potential <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>ν</mi></mrow></msub><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>ν</mi></mrow></msub></math></span> where, for <span><math><mi>b</mi><mo>∈</mo><mi>R</mi></math></span> and <span><math><mi>ν</mi><mo>∈</mo><mi>R</mi></math></span>, <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>b</mi><mspace></mspace><mo>(</mo><mo>−</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is the Aharonov-Bohm potential centered at the origin of flux <span><math><mn>2</mn><mi>π</mi><mi>ν</mi></math></span>. First, we show that the limit of <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>(</mo><mi>b</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> as <span><math><mi>b</mi><mo>→</mo><mn>0</mn></math></span> is equal to the D-to-N map <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>ν</mi><mo>)</mo></math></span> on the interior of the disk associated with the potential <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span>. Secondly, we study the ground state energy of the D-to-N map <span><math><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover><mo>(</mo><mi>b</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results as <span><math><mi>b</mi><mo>→</mo><mo>∞</mo></math></span> obtained in the interior case for general domains.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103799"},"PeriodicalIF":2.3,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219206","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":"Monotonicity formulas for capillary surfaces","authors":"Guofang Wang ,&nbsp;Chao Xia ,&nbsp;Xuwen Zhang","doi":"10.1016/j.matpur.2025.103802","DOIUrl":"10.1016/j.matpur.2025.103802","url":null,"abstract":"<div><div>In this paper, we establish monotonicity formulas for capillary surfaces in the half-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> and in the unit ball <span><math><msup><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and extend the result of Volkmann (2016) <span><span>[27]</span></span> for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in <span><math><msup><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> (2011) <span><span>[10]</span></span> to the capillary setting, which is different to another optimal area estimate proved by Brendle (2023) <span><span>[5]</span></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103802"},"PeriodicalIF":2.3,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219417","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 long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions","authors":"Deqin Zhou ,&nbsp;Felipe Linares","doi":"10.1016/j.matpur.2025.103792","DOIUrl":"10.1016/j.matpur.2025.103792","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We consider the long time behavior of the solutions of the coupled Schrödinger-KdV system&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; We show that global solutions to this system satisfy locally energy decay in a suitable interval, growing unbounded in time, in two situations. In the first case, we regard the parameter vector &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; without any size assumption on the initial data in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In the second one, we consider the parameter vector &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. In this case, we give a ‘‘smallness” criterion involving the product of the parameter −&lt;em&gt;β&lt;/em&gt; and a constant depending on the initial data in &lt;span&gt;&lt;","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103792"},"PeriodicalIF":2.3,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145094983","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}