Journal of Differential Equations最新文献

{"title":"Energy conservation for compressible fluid systems with Korteweg stress tensors","authors":"Guilong Gui ,&nbsp;Tong Tang","doi":"10.1016/j.jde.2025.02.029","DOIUrl":"10.1016/j.jde.2025.02.029","url":null,"abstract":"<div><div>Energy conservation is an important issue in Onsager's conjecture. We consider in the paper the weak solutions of compressible quantum Euler system and quantum Navier-Stokes system under what regularity conditions conserve the energy. Based on the work of Bresch et al. (2019) <span><span>[8]</span></span> and Feireisl et al. (2017) <span><span>[20]</span></span>, we introduce the drift velocity and the effective velocity to write the two quantum fluid systems and obtain the corresponding augmented systems as the compressible Navier-Stokes system with density dependent viscosity, then prove the energy conservation for the augmented system, which eliminate the third order dispersive term. We find some new observations and phenomena, which is different from the previous results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 330-347"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Systems of differential equations of higher order describing pseudo-spherical or spherical surfaces","authors":"Filipe Kelmer ,&nbsp;Keti Tenenblat","doi":"10.1016/j.jde.2025.02.003","DOIUrl":"10.1016/j.jde.2025.02.003","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We consider systems of partial differential equations of the form&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&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;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;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&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;mi&gt;v&lt;/mi&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;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&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;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&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;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;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&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;mi&gt;v&lt;/mi&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;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, describing pseudospherical (&lt;strong&gt;pss&lt;/strong&gt;) or spherical surfaces (&lt;strong&gt;ss&lt;/strong&gt;). Generic solutions of such a system define metrics on open subsets of the plane, with coordinates &lt;span&gt;&lt;math&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;/math&gt;&lt;/span&gt;, whose Gaussian curvature is &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. These systems are the integrability conditions of &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-valued linear problems, with &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;sl&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&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; or &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;su&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, when &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, respectively. We obtain classification results for classes of such systems of differential equations of order &lt;em&gt;n&lt;/em&gt; and &lt;em&gt;m&lt;/em&gt;, in terms of four arbitrary smooth functions satisfying certain generic conditions. We also provide classification results for special type of second and third order systems. We include several explicit examples. Applications of the theory provide new examples and new families of","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"424 ","pages":"Pages 833-858"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Zero-Mach limit of the compressible Navier-Stokes system on 2D exterior domains with non-slip boundary conditions for all time","authors":"Xinyu Fan ,&nbsp;Qiangchang Ju ,&nbsp;Zilai Li ,&nbsp;Jianjun Xu","doi":"10.1016/j.jde.2025.02.030","DOIUrl":"10.1016/j.jde.2025.02.030","url":null,"abstract":"<div><div>We prove the zero-Mach limit of the compressible Navier-Stokes system on 2D exterior domains with non-slip boundary conditions for all time. Compared with the previous works on the domain with no boundary or slip boundary conditions, the non-slip conditions generate essential difficulties in obtaining uniform near-boundary estimates of smooth solutions. The key ingredient of our work includes the derivation of uniform estimates for the high order derivatives of the density from the hyperbolic dissipations. Moreover, we develop some new global geometric tools based on the decomposition of Euclidean metric to handle the tough boundary estimates, and the method seems applicable for other boundary problems on exterior domains as well.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 291-329"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: A Levinson type theorem on complex domains, and applications","authors":"Giordano Cotti ,&nbsp;Davide Guzzetti ,&nbsp;Davide Masoero","doi":"10.1016/j.jde.2025.01.085","DOIUrl":"10.1016/j.jde.2025.01.085","url":null,"abstract":"<div><div>In this paper, we consider systems of linear ordinary differential equations, with analytic coefficients on big sectorial domains, which are asymptotically diagonal for large values of <span><math><mo>|</mo><mi>z</mi><mo>|</mo></math></span>. Inspired by <span><span>[60]</span></span>, we introduce two conditions on the dominant diagonal term (the <em>L</em>-<em>condition</em>) and on the perturbation term (the <em>good decay condition</em>) of the coefficients of the system, respectively. Assuming the validity of these conditions, we then show the existence and uniqueness, on big sectorial domains, of an <em>asymptotic</em> fundamental matrix solution, i.e. asymptotically equivalent (for large <span><math><mo>|</mo><mi>z</mi><mo>|</mo></math></span>) to a fundamental system of solutions of the unperturbed diagonal system. Moreover, a refinement (in the case of subdominant solutions) and a generalization (in the case of systems depending on parameters) of this result are given.</div><div>As a first application, we address the study of a class of ODEs with not-necessarily meromorphic coefficients, the leading diagonal term of the coefficient being a generalized polynomial in <em>z</em> with real exponents. We provide sufficient conditions on the coefficients ensuring the existence and uniqueness of an asymptotic fundamental system of solutions, and we give an explicit description of the maximal sectors of validity for such an asymptotics. Furthermore, we also focus on distinguished examples in this class of ODEs arising in the context of open conjectures in Mathematical Physics relating Integrable Quantum Field Theories and affine opers (<em>ODE/IM correspondence</em>). Notably, our results fill two significant gaps in the mathematical literature pertaining to these conjectural relations.</div><div>Finally, as a second application, we consider the classical case of ODEs with meromorphic coefficients. Under an <em>adequateness</em> condition on the coefficients (allowing ramification of the irregular singularities), we show that our results reproduce (with a shorter proof) the main asymptotic existence theorems of Y. Sibuya <span><span>[80]</span></span>, <span><span>[81]</span></span> and W. Wasow <span><span>[94]</span></span> in their optimal refinements: the sectors of validity of the asymptotics are maximal, and the asymptotic fundamental system of solutions is unique.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 1-58"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global regularity and time decay for the SQG equation with anisotropic fractional dissipation","authors":"Zhuan Ye","doi":"10.1016/j.jde.2025.02.022","DOIUrl":"10.1016/j.jde.2025.02.022","url":null,"abstract":"<div><div>In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. On the one hand, when the dissipation powers are restricted to a suitable range, the global regularity of the surface quasi-geostrophic equation is obtained by some anisotropic embedding and interpolation inequalities involving fractional derivatives. On the other hand, we obtain the optimal large time decay estimates for global weak solutions by an anisotropic interpolation inequality. Moreover, based on the argument adopted in establishing the global <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm of the solution, we obtain the optimal large time decay estimates for the above obtained global smooth solutions. Finally, the decay estimates for the difference between the full solution and the solution to the corresponding linear part are also derived.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 159-229"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence and convergence of stochastic processes underlying a thin layer approximation of a coupled bulk-surface PDE","authors":"Adam Bobrowski ,&nbsp;Anotida Madzvamuse ,&nbsp;Elżbieta Ratajczyk","doi":"10.1016/j.jde.2025.02.011","DOIUrl":"10.1016/j.jde.2025.02.011","url":null,"abstract":"<div><div>We study a system of coupled bulk-surface partial differential equations (BS-PDEs), describing changes in concentration of certain proteins (Rho GTPases) in a living cell. These proteins, when activated, are bound to the plasma membrane where they diffuse and react with the inactive species; inactivated species diffuse inside the cell cortex; these react with the activated species when they are close to the plasma membrane. For our case study, we model the cell cortex as an annulus, and the plasma membrane as its outer circle.</div><div>Mathematically, the aim of the paper is twofold: Firstly, we show the master equation for the changes in concentration of Rho GTPases is the Kolmogorov forward differential equation for an underlying Feller stochastic process, and, in particular, the related Cauchy problem is well-posed. Secondly, since the cell cortex is typically a rather thin domain, we study the situation where the thickness of the annulus modeling the cortex converges to 0. To this end, we note that letting the thickness of the annulus to 0 is equivalent to keeping it constant while increasing the rate of radial diffusion. As a result, in the limit, solutions to the master equation lose dependence on the radial coordinate and can be thought of as functions on the circle. Furthermore, the limit master equation can be seen as describing diffusion on two copies of the circle with jumps from one copy to the other.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 113-158"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Traveling waves of the Vlasov–Poisson system","authors":"Masahiro Suzuki ,&nbsp;Masahiro Takayama ,&nbsp;Katherine Zhiyuan Zhang","doi":"10.1016/j.jde.2025.02.021","DOIUrl":"10.1016/j.jde.2025.02.021","url":null,"abstract":"<div><div>We consider the Vlasov–Poisson system describing a two-species plasma with spatial dimension 1 and the velocity variable in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We find the necessary and sufficient conditions for the existence of solitary waves, shock waves, and wave trains of the system, respectively. To this end, we need to investigate solutions that are not BGK waves. Furthermore, we classify completely in all possible cases whether or not the traveling wave is unique. The uniqueness varies according to each traveling wave when we exclude the variant caused by translation. For the solitary wave, there are both cases that it is unique and nonunique. The shock wave is always unique. No wave train is unique.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 230-290"},"PeriodicalIF":2.4,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiple boundary peak solution for critical elliptic system with Neumann boundary condition","authors":"Yuxia Guo,&nbsp;Shengyu Wu,&nbsp;Tingfeng Yuan","doi":"10.1016/j.jde.2025.02.015","DOIUrl":"10.1016/j.jde.2025.02.015","url":null,"abstract":"<div><div>We consider the following elliptic system with Neumann boundary condition:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>μ</mi><mi>u</mi><mo>=</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>v</mi><mo>+</mo><mi>μ</mi><mi>v</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>v</mi></mrow><mrow><mo>∂</mo><mi>n</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>on </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>v</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is a smooth bounded domain, <em>μ</em> is a positive constant and <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> lies in the critical hyperbola:<span><span><span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mo>.</mo></math></span></span></span> By using the Lyapunov-Schmidt reduction technique, we establish the existence of infinitely many solutions to above system. These solutions have multiple peaks that are located on the boundary ∂Ω. Our results show that the geometry of the boundary ∂Ω, especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"428 ","pages":"Pages 59-112"},"PeriodicalIF":2.4,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamical localization for finitely differentiable quasi-periodic long-range operators","authors":"Yuan Shan","doi":"10.1016/j.jde.2025.02.012","DOIUrl":"10.1016/j.jde.2025.02.012","url":null,"abstract":"<div><div>In this paper, we establish the <em>s</em>-power law dynamical localization for a class of finitely differentiable quasi-periodic long-range operators on <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> with Diophantine frequencies. This result represents the strongest form of dynamical localization in the setting of finitely differentiable topology which is a generalization of exponential dynamical localization in expectation in the analytic case. Our approach is based on the Aubry duality and quantitative reducibility theorem of the finitely differentiable <span><math><mtext>SL</mtext><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></math></span> quasi-periodic cocycles in the local regime. The <em>s</em>-power law dynamical localization discussed here also demonstrates strong ballistic transport for finitely differentiable quasi-periodic Schrödinger operators.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 803-826"},"PeriodicalIF":2.4,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Axial symmetry and refined spike profiles of ground states for rotating two-component Bose gases","authors":"Yongshuai Gao ,&nbsp;Yong Luo","doi":"10.1016/j.jde.2025.02.017","DOIUrl":"10.1016/j.jde.2025.02.017","url":null,"abstract":"<div><div>This paper considers ground states of two-component Bose gases confined in an anharmonic trap rotating at the velocity <span><math><mn>0</mn><mo>&lt;</mo><mi>Ω</mi><mo>&lt;</mo><mo>∞</mo></math></span>, where the intraspecies interaction <span><math><mo>(</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> and the interspecies interaction −<em>β</em> are both attractive, i.e., <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <em>β</em> are all positive. We prove the axially symmetry and the refined spike profiles of ground states as <span><math><mi>β</mi><mo>↗</mo><msup><mrow><mi>β</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>:</mo><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>+</mo><msqrt><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></msqrt></math></span>, where <span><math><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>:</mo><mo>=</mo><msubsup><mrow><mo>‖</mo><mi>Q</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> are fixed and <span><math><mi>Q</mi><mo>&gt;</mo><mn>0</mn></math></span> is the unique positive solution of <span><math><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>u</mi><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mn>0</mn></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 761-802"},"PeriodicalIF":2.4,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143377053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}