Journal of Differential Equations最新文献

{"title":"On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D","authors":"Guangyi Hong ,&nbsp;Tao Luo ,&nbsp;Zhonghao Zhao","doi":"10.1016/j.jde.2024.11.026","DOIUrl":"10.1016/j.jde.2024.11.026","url":null,"abstract":"<div><div>In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in <span><span>[14, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2019]</span></span> from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 40-80"},"PeriodicalIF":2.4,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722411","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":"Global analyticity and the lower bounds of analytic radius for the Chaplygin gas equations with source terms","authors":"Zhengyan Liu ,&nbsp;Xinglong Wu ,&nbsp;Boling Guo","doi":"10.1016/j.jde.2024.11.027","DOIUrl":"10.1016/j.jde.2024.11.027","url":null,"abstract":"<div><div>This paper is devoted to studying the global existence and the analytic radius of analytic solutions to the Chaplygin gas equations with source terms. If the initial data belongs to Gevrey spaces and it is sufficiently small, we show the solution has the global persistent property in Gevrey spaces. In particular, we obtain uniform lower bounds on the spatial analytic radius which is given by <span><math><mi>C</mi><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>C</mi><mi>t</mi></mrow></msup></math></span>, for some constant <span><math><mi>C</mi><mo>&gt;</mo><mn>0</mn></math></span>, this tells us that the decay rate of the analytic radius is at most a single exponential decay, which is the slowest decay rate of lower bounds on the analytic radius compared with the double and triple exponential decay of analytic radius derived by Levermore, Bardos, et al. (see <span><span>Remark 1.2</span></span>). Our method is based on the Fourier transformation and Gevrey-class regularity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 81-113"},"PeriodicalIF":2.4,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722412","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":"A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space","authors":"Monideep Ghosh,&nbsp;Anumol Joseph,&nbsp;Debabrata Karmakar","doi":"10.1016/j.jde.2024.11.025","DOIUrl":"10.1016/j.jde.2024.11.025","url":null,"abstract":"<div><div>We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space<span><span><span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><msup><mrow><mi>B</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></msub><mi>u</mi><mo>+</mo><mi>λ</mi><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><mi>θ</mi><mi>u</mi><mi>ln</mi><mo>⁡</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><mi>u</mi><mo>&gt;</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>B</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span> and study the existence vs non-existence results. We show that whenever <span><math><mi>θ</mi><mo>&gt;</mo><mn>0</mn></math></span>, there exists an <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-solution, while for <span><math><mi>θ</mi><mo>&lt;</mo><mn>0</mn></math></span>, there does not exist a positive solution in a reasonably general class. Since the perturbation <span><math><mi>u</mi><mi>ln</mi><mo>⁡</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> changes sign, Pohozaev type identities do not yield any non-existence results. The main contribution of this article is obtaining an “almost” precise lower asymptotic decay estimate on the positive solutions for <span><math><mi>θ</mi><mo>&lt;</mo><mn>0</mn></math></span>, culminating in proving their non-existence assertion.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 114-149"},"PeriodicalIF":2.4,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746148","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":"Time splitting method for nonlinear Schrödinger equation with rough initial data in L2","authors":"Hyung Jun Choi ,&nbsp;Seonghak Kim ,&nbsp;Youngwoo Koh","doi":"10.1016/j.jde.2024.11.018","DOIUrl":"10.1016/j.jde.2024.11.018","url":null,"abstract":"<div><div>We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>,<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>i</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>λ</mi><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace></mtd><mtd><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>λ</mi><mo>∈</mo><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> and <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span>. While the Lie approximation <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span> is known to converge to the solution <em>u</em> when the initial datum <em>ϕ</em> is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data <span><math><mi>ϕ</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>, we prove the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> convergence of the filtered Lie approximation <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>f</mi><mi>l</mi><mi>t</mi></mrow></msub></math></span> to the solution <em>u</em> in the mass-subcritical range, <span><math><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>d</mi></mrow></mfrac></math></span>. Furthermore, we provide a precise convergence result for radial initial data <span><math><mi>ϕ</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"417 ","pages":"Pages 164-190"},"PeriodicalIF":2.4,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720436","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":"The extended second Painlevé hierarchy: Auto-Bäcklund transformations and special integrals","authors":"P.R. Gordoa,&nbsp;A. Pickering","doi":"10.1016/j.jde.2024.11.017","DOIUrl":"10.1016/j.jde.2024.11.017","url":null,"abstract":"<div><div>We return to our study of the extended second Painlevé hierarchy presented in a previous paper. For this hierarchy we give a new local auto-BT. We also give an extensive discussion of the iterative construction of solutions and special integrals using auto-BTs. Furthermore, we show that Lax pairs can be provided for special integrals. Even though this will, in fact, be the case quite generally, it seems that Lax pairs for special integrals have not been given previously. Amongst the equations for which we present Lax pairs are examples due to Cosgrove and, in classical Painlevé classification results, Chazy.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"417 ","pages":"Pages 132-163"},"PeriodicalIF":2.4,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720435","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":"Asymptotically additive families of functions and a physical equivalence problem for flows","authors":"Carllos Eduardo Holanda","doi":"10.1016/j.jde.2024.11.023","DOIUrl":"10.1016/j.jde.2024.11.023","url":null,"abstract":"<div><div>We show that additive and asymptotically additive families of continuous functions with respect to suspension flows are physically equivalent. In particular, the equivalence result holds for hyperbolic flows. We also obtain an equivalence relation for expansive flows. Moreover, we show how this equivalence result can be used to extend the nonadditive thermodynamic formalism and multifractal analysis for flows.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 142-177"},"PeriodicalIF":2.4,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723304","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":"Global well-posedness of the three-dimensional free boundary problem for viscoelastic fluids without surface tension","authors":"Jingchi Huang,&nbsp;Zheng-an Yao,&nbsp;Xiangyu You","doi":"10.1016/j.jde.2024.11.020","DOIUrl":"10.1016/j.jde.2024.11.020","url":null,"abstract":"<div><div>In this paper, we consider the three-dimensional free boundary problem of incompressible and compressible neo-Hookean viscoelastic fluid equations in an infinite strip without surface tension, provided that the initial data is sufficiently close to the equilibrium state. By reformulating the problems in Lagrangian coordinates, we can get the stabilizing effect of elasticity. In both cases, we utilize the elliptic estimates to improve the estimates. Moreover, for the compressible case, we find there is an extra ODE structure that can improve the regularity of the free boundary, thus we can have the global well-posedness. To prove the global well-posedness for the incompressible case, we employ two-tier energy method introduced in <span><span>[11]</span></span><span><span>[12]</span></span><span><span>[13]</span></span> to compensate for the inferior structure.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"417 ","pages":"Pages 191-230"},"PeriodicalIF":2.4,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720437","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":"Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions","authors":"H. Bustos ,&nbsp;P. Figueroa ,&nbsp;M. Pinto","doi":"10.1016/j.jde.2024.11.016","DOIUrl":"10.1016/j.jde.2024.11.016","url":null,"abstract":"<div><div>We address the Poincaré-Perron's classical problem of approximation for <em>high order</em> linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in <span><span>[23]</span></span>. We obtain explicit formulae for solutions of these equations, for any fixed order <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also <em>p</em>-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"417 ","pages":"Pages 231-259"},"PeriodicalIF":2.4,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721006","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":"Explicit approximation for stochastic nonlinear Schrödinger equation","authors":"Jianbo Cui","doi":"10.1016/j.jde.2024.11.022","DOIUrl":"10.1016/j.jde.2024.11.022","url":null,"abstract":"<div><div>In this paper, we study explicit approximations of stochastic nonlinear Schrödinger equations (SNLSEs). We first prove that the classical explicit numerical approximations are divergent for SNLSEs with polynomial nonlinearities. To enhance the stability, we propose a kind of explicit numerical approximations, and establish the regularity analysis and strong convergence rate of the proposed approximations for SNLSEs. There are two key ingredients in our approach. One ingredient is constructing a logarithmic auxiliary functional and exploiting the Bourgain space to prove new regularity estimates of SNLSEs. Another one is providing a dedicated error decomposition formula and presenting the tail estimates of underlying stochastic processes. In particular, our result answers the strong convergence problem of numerical approximation for 2D SNLSEs.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 1-39"},"PeriodicalIF":2.4,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701154","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":"Corrigendum to “Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework” [J. Differ. Equ. 271 (2021) 594–637]","authors":"Benoît Bonnet-Weill ,&nbsp;Hélène Frankowska","doi":"10.1016/j.jde.2024.10.045","DOIUrl":"10.1016/j.jde.2024.10.045","url":null,"abstract":"<div><div>This corrigendum is concerned with the technical preliminary <span><span>[1, Lemma 1]</span></span>. Unfortunately, its proof contains a mistake which ultimately renders its conclusion erroneous. In this note, we provide a corrected version of the latter, and show that this modification has no impact on the other results of <span><span>[1]</span></span> while incurring very benign changes in none but two series of computations throughout the manuscript.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2324-2327"},"PeriodicalIF":2.4,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744165","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}