Pub Date : 2026-01-16DOI: 10.1016/j.jde.2026.114110
Ya-Guang Wang , Yi-Lei Zhao
We study the well-posedness of the compressible boundary layer equations with data being analytic in the tangential variable of the boundary. The compressible boundary layer equations, a nonlinear coupled system of degenerate parabolic equations and an elliptic equation, describe the behavior of thermal layer and viscous layer in the small viscosity and heat conductivity limit, for the two-dimensional compressible viscous flow with heat conduction with nonslip and zero heat flux boundary conditions. We use the Littlewood-Paley theory to establish the a priori estimates for solutions of this compressible boundary layer problem, and obtain the local existence and uniqueness of the solution in the space of analytic in the tangential variable and Sobolev in the normal variable.
{"title":"Well-posedness of the compressible boundary layer equations with analytic initial data","authors":"Ya-Guang Wang , Yi-Lei Zhao","doi":"10.1016/j.jde.2026.114110","DOIUrl":"10.1016/j.jde.2026.114110","url":null,"abstract":"<div><div>We study the well-posedness of the compressible boundary layer equations with data being analytic in the tangential variable of the boundary. The compressible boundary layer equations, a nonlinear coupled system of degenerate parabolic equations and an elliptic equation, describe the behavior of thermal layer and viscous layer in the small viscosity and heat conductivity limit, for the two-dimensional compressible viscous flow with heat conduction with nonslip and zero heat flux boundary conditions. We use the Littlewood-Paley theory to establish the a priori estimates for solutions of this compressible boundary layer problem, and obtain the local existence and uniqueness of the solution in the space of analytic in the tangential variable and Sobolev in the normal variable.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114110"},"PeriodicalIF":2.3,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980609","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}
Pub Date : 2026-01-16DOI: 10.1016/j.jde.2026.114113
Ryuji Kajikiya
We study the Hénon equation in unbounded domains Ω which are G invariant, where G is a closed subgroup of the orthogonal group. We say that Ω (or ) is G invariant if (or ) for any . We call a least energy solution if it is a minimizer of the Rayleigh quotient associated with the Hénon equation. We offer sufficient conditions which guarantee that no least energy solution is G invariant.
{"title":"Group noninvariant solutions of the Hénon equation in unbounded domains","authors":"Ryuji Kajikiya","doi":"10.1016/j.jde.2026.114113","DOIUrl":"10.1016/j.jde.2026.114113","url":null,"abstract":"<div><div>We study the Hénon equation in unbounded domains Ω which are <em>G</em> invariant, where <em>G</em> is a closed subgroup of the orthogonal group. We say that Ω (or <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>) is <em>G</em> invariant if <span><math><mi>g</mi><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>=</mo><mi>Ω</mi></math></span> (or <span><math><mi>u</mi><mo>(</mo><mi>g</mi><mi>x</mi><mo>)</mo><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>) for any <span><math><mi>g</mi><mo>∈</mo><mi>G</mi></math></span>. We call <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> a least energy solution if it is a minimizer of the Rayleigh quotient associated with the Hénon equation. We offer sufficient conditions which guarantee that no least energy solution is <em>G</em> invariant.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114113"},"PeriodicalIF":2.3,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980608","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}
Pub Date : 2026-01-15DOI: 10.1016/j.jde.2026.114109
Paulo L. Dattori da Silva, André Pedroso Kowacs
We extend the directional Poincaré inequality on the torus, introduced by Steinerberger in (2016) [1], to the setting of compact Lie groups. We provide necessary and sufficient conditions for the existence of such an inequality based on estimates on the eigenvalues of the global symbol of the corresponding vector field. We also prove that such refinement of the Poincaré inequality holds for a left-invariant vector field on a compact Lie group G if and only if the vector field is globally solvable, and extend this equivalence to tube-type vector fields on .
{"title":"Directional Poincaré inequality on compact Lie groups","authors":"Paulo L. Dattori da Silva, André Pedroso Kowacs","doi":"10.1016/j.jde.2026.114109","DOIUrl":"10.1016/j.jde.2026.114109","url":null,"abstract":"<div><div>We extend the directional Poincaré inequality on the torus, introduced by Steinerberger in (2016) <span><span>[1]</span></span>, to the setting of compact Lie groups. We provide necessary and sufficient conditions for the existence of such an inequality based on estimates on the eigenvalues of the global symbol of the corresponding vector field. We also prove that such refinement of the Poincaré inequality holds for a left-invariant vector field on a compact Lie group <em>G</em> if and only if the vector field is globally solvable, and extend this equivalence to tube-type vector fields on <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><mi>G</mi></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114109"},"PeriodicalIF":2.3,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980610","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2025.114081
Ziyue Zeng, Yuxiang Li
<div><div>This paper investigates the quasilinear two-species chemotaxis system with two chemicals<span><span><span>(⋆)</span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>∇</mi><mo>⋅</mo><mrow><mo>(</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>v</mi><mo>)</mo></mrow><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mo>⨏</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>∇</mi><mo>⋅</mo><mrow><mo>(</mo><mi>g</mi><mo>(</mo><mi>w</mi><mo>)</mo><mi>∇</mi><mi>z</mi><mo>)</mo></mrow><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>z</mi><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mo>⨏</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>v</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>w</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>z</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>w</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></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> <span><math><mo>(</mo><mi>n</mi><mo>⩾</mo><mn>3</mn><mo>)</mo></math></span> is a smoothly bounded domain. The sensitivity functi
{"title":"Critical blow-up curve in a quasilinear two-species chemotaxis system with two chemicals","authors":"Ziyue Zeng, Yuxiang Li","doi":"10.1016/j.jde.2025.114081","DOIUrl":"10.1016/j.jde.2025.114081","url":null,"abstract":"<div><div>This paper investigates the quasilinear two-species chemotaxis system with two chemicals<span><span><span>(⋆)</span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>∇</mi><mo>⋅</mo><mrow><mo>(</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>v</mi><mo>)</mo></mrow><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mo>⨏</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>∇</mi><mo>⋅</mo><mrow><mo>(</mo><mi>g</mi><mo>(</mo><mi>w</mi><mo>)</mo><mi>∇</mi><mi>z</mi><mo>)</mo></mrow><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>z</mi><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mo>⨏</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>v</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>w</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>z</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>w</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></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> <span><math><mo>(</mo><mi>n</mi><mo>⩾</mo><mn>3</mn><mo>)</mo></math></span> is a smoothly bounded domain. The sensitivity functi","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114081"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977988","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2025.114091
Jing Gao , Weijun Zhang
<div><div>In this paper, we investigate bifurcation phenomena for fully nonlinear elliptic equations and coupled systems dominated by <em>k</em>-Hessian operator. Specifically, we consider the Dirichlet problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>f</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span></div><div>as well as the coupled system<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>g</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>v</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>h</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>v</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span> where Ω is a bounded strictly (<em>k</em>-1)-convex domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>λ</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><mi>f</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> is a continuous function with zeros only at 0 and <span><math><mi>g</mi><mo>,</mo><mi>h</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>×</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> are continuous functions with zeros only at <span><math><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> and <span><math><mo
{"title":"Bifurcation on fully nonlinear elliptic equations and systems","authors":"Jing Gao , Weijun Zhang","doi":"10.1016/j.jde.2025.114091","DOIUrl":"10.1016/j.jde.2025.114091","url":null,"abstract":"<div><div>In this paper, we investigate bifurcation phenomena for fully nonlinear elliptic equations and coupled systems dominated by <em>k</em>-Hessian operator. Specifically, we consider the Dirichlet problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>f</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span></div><div>as well as the coupled system<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>g</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>v</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>h</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>v</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span> where Ω is a bounded strictly (<em>k</em>-1)-convex domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>λ</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><mi>f</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> is a continuous function with zeros only at 0 and <span><math><mi>g</mi><mo>,</mo><mi>h</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>×</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> are continuous functions with zeros only at <span><math><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> and <span><math><mo","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114091"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978284","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2026.114106
Divyang G. Bhimani , Diksha Dhingra , Vijay Kumar Sohani
The study of low regularity Cauchy data for nonlinear dispersive PDEs has been successfully achieved using modulation spaces in recent years. In this paper, we study the inhomogeneous nonlinear Schrödinger equation (INLS) on the whole space having initial data in modulation spaces. In the subcritical regime , we establish local well-posedness in . By adapting Bourgain's high-low decomposition method, we establish global well-posedness in with and p sufficiently close to 2. This is the first global well-posedness result for INLS in modulation spaces, which contains certain Sobolev and Sobolev spaces.
{"title":"Low-regularity global solution of the inhomogeneous nonlinear Schrödinger equations in modulation spaces","authors":"Divyang G. Bhimani , Diksha Dhingra , Vijay Kumar Sohani","doi":"10.1016/j.jde.2026.114106","DOIUrl":"10.1016/j.jde.2026.114106","url":null,"abstract":"<div><div>The study of low regularity Cauchy data for nonlinear dispersive PDEs has been successfully achieved using modulation spaces in recent years. In this paper, we study the inhomogeneous nonlinear Schrödinger equation (INLS)<span><span><span><math><mi>i</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>Δ</mi><mi>u</mi><mo>±</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>b</mi></mrow></msup><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>α</mi></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mspace></mspace><mo>(</mo><mi>b</mi><mo>,</mo><mi>α</mi><mo>></mo><mn>0</mn><mo>)</mo></math></span></span></span> on the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> having initial data in modulation spaces. In the subcritical regime <span><math><mo>(</mo><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mn>2</mn><mi>b</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></math></span>, we establish local well-posedness in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>2</mn><mo>,</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>(</mo><mo>⊃</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mspace></mspace><mtext>for</mtext><mspace></mspace><mi>s</mi><mo>></mo><mfrac><mrow><mi>n</mi><mi>α</mi></mrow><mrow><mn>2</mn><mo>(</mo><mi>α</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mfrac><mo>)</mo></math></span>. By adapting Bourgain's high-low decomposition method, we establish global well-posedness in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>p</mi><mo>,</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></math></span> with <span><math><mn>2</mn><mo><</mo><mi>p</mi></math></span> and <em>p</em> sufficiently close to 2. This is the first global well-posedness result for INLS in modulation spaces, which contains certain Sobolev <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> <span><math><mo>(</mo><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>−</mo></math></span>Sobolev spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114106"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977526","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2026.114108
Zijin Li , Ning Liu , Taoran Zhou
We consider the generalized Leray's problem with the Navier-slip boundary condition in an infinite pipe . We show that if the flux Φ of the solution is no larger than a critical value that is independent with the friction ratio of the Navier-slip boundary condition, the solution to the problem must be the parallel Poiseuille flow with the given flux. Compared with known related 3D results, this seems to be the first conclusion with the size of critical flux being uniform with the friction ratio , and it is surprising since the prescribed uniqueness breaks down immediately when , even if .
Our proof relies primarily on a refined gradient estimate of the Poiseuille flow with the Navier-slip boundary condition. Additionally, we prove the critical flux provided that Σ is a unit disk.
{"title":"A refined uniqueness result of Leray's problem in an infinite-long pipe with the Navier-slip boundary","authors":"Zijin Li , Ning Liu , Taoran Zhou","doi":"10.1016/j.jde.2026.114108","DOIUrl":"10.1016/j.jde.2026.114108","url":null,"abstract":"<div><div>We consider the generalized Leray's problem with the Navier-slip boundary condition in an infinite pipe <span><math><mi>D</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mi>R</mi></math></span>. We show that if the flux Φ of the solution is no larger than a critical value that is <em>independent with the friction ratio</em> of the Navier-slip boundary condition, the solution to the problem must be the parallel Poiseuille flow with the given flux. Compared with known related 3D results, this seems to be the first conclusion with the size of critical flux being uniform with the friction ratio <span><math><mi>α</mi><mo>∈</mo><mo>]</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>]</mo></math></span>, and it is surprising since the prescribed uniqueness breaks down immediately when <span><math><mi>α</mi><mo>=</mo><mn>0</mn></math></span>, even if <span><math><mi>Φ</mi><mo>=</mo><mn>0</mn></math></span>.</div><div>Our proof relies primarily on a refined gradient estimate of the Poiseuille flow with the Navier-slip boundary condition. Additionally, we prove the critical flux <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>16</mn></mrow></mfrac></math></span> provided that Σ is a unit disk.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114108"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980653","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2026.114104
Chuchu Chen , Tonghe Dang , Jialin Hong , Guoting Song
This paper studies densities for solutions of the stochastic functional differential equation (SFDE) and of its Euler-type discretizations. First, by means of the Malliavin calculus, we prove the existence of densities for the exact solution and its discretizations. Then we establish the -convergence for the density of discretizations by implementing a dimensionality reduction argument and a localization argument. Further, we prove that the pointwise convergence rate of the density is 1 when the noise is of additive type. The convergence results indicate that the total variation distance between laws of solutions for the SFDE and its discretizations vanishes to zero as the discretization parameter diminishes, while that between laws of functional solutions fails to vanish due to the high degeneracy of the equation. This finding highlights one of the main distinctions in asymptotic behaviors of the corresponding discretized systems when compared to stochastic ordinary (partial) differential equations.
{"title":"Densities of stochastic functional differential equation and its discretizations","authors":"Chuchu Chen , Tonghe Dang , Jialin Hong , Guoting Song","doi":"10.1016/j.jde.2026.114104","DOIUrl":"10.1016/j.jde.2026.114104","url":null,"abstract":"<div><div>This paper studies densities for solutions of the stochastic functional differential equation (SFDE) and of its Euler-type discretizations. First, by means of the Malliavin calculus, we prove the existence of densities for the exact solution and its discretizations. Then we establish the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>-convergence for the density of discretizations by implementing a dimensionality reduction argument and a localization argument. Further, we prove that the pointwise convergence rate of the density is 1 when the noise is of additive type. The convergence results indicate that the total variation distance between laws of solutions for the SFDE and its discretizations vanishes to zero as the discretization parameter diminishes, while that between laws of functional solutions fails to vanish due to the high degeneracy of the equation. This finding highlights one of the main distinctions in asymptotic behaviors of the corresponding discretized systems when compared to stochastic ordinary (partial) differential equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114104"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978281","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2026.114099
Li-Ming Cao, Shou-Fu Tian
The main purpose of this work is to investigate the spectral stability of elliptic function solutions for the short pulse equation, a completely integrable model for the description of ultra-short pulse propagation in optical fibers. Recently, Yang and Fan developed -steepest descent method to analyze the long-time asymptotic behavior for the short pulse equation (Yang and Fan (2021) [50]). Subsequently, Li, Tian and Yang extended their results and reported the asymptotic stability of N-soliton solution (Li et al. (2023) [33]). Inspired by these works, we consider the spectral stability for three classes of elliptic solutions which are derived via the algebraic geometry method in this work. It is worth noting that the Lax spectrum in focusing case is not restricted to imaginary axis. To address this issue, we develop the squared wavefunction method using Jacobi theta function theory, and then establish spectral stability for both focusing and defocusing cases.
本文的主要目的是研究用于描述超短脉冲在光纤中传播的完全可积模型——短脉冲方程的椭圆函数解的谱稳定性。最近，Yang和Fan开发了∂¯-最陡下降方法来分析短脉冲方程的长时间渐近行为（Yang和Fan(2021)[50]）。随后，Li、Tian和Yang扩展了他们的结果，报道了n孤子解的渐近稳定性（Li et al.(2023)[33]）。受这些工作的启发，本文研究了用代数几何方法导出的三类椭圆解的谱稳定性。值得注意的是，聚焦情况下的Lax光谱并不局限于虚轴。为了解决这个问题，我们利用Jacobi theta函数理论开发了平方波函数方法，然后建立了聚焦和散焦情况下的光谱稳定性。
{"title":"Spectral stability of elliptic function solutions for the short pulse equation","authors":"Li-Ming Cao, Shou-Fu Tian","doi":"10.1016/j.jde.2026.114099","DOIUrl":"10.1016/j.jde.2026.114099","url":null,"abstract":"<div><div>The main purpose of this work is to investigate the spectral stability of elliptic function solutions for the short pulse equation, a completely integrable model for the description of ultra-short pulse propagation in optical fibers. Recently, Yang and Fan developed <span><math><mover><mrow><mo>∂</mo></mrow><mrow><mo>¯</mo></mrow></mover></math></span>-steepest descent method to analyze the long-time asymptotic behavior for the short pulse equation (Yang and Fan (2021) <span><span>[50]</span></span>). Subsequently, Li, Tian and Yang extended their results and reported the asymptotic stability of N-soliton solution (Li et al. (2023) <span><span>[33]</span></span>). Inspired by these works, we consider the spectral stability for three classes of elliptic solutions which are derived via the algebraic geometry method in this work. It is worth noting that the Lax spectrum in focusing case is not restricted to imaginary axis. To address this issue, we develop the squared wavefunction method using Jacobi theta function theory, and then establish spectral stability for both focusing and defocusing cases.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114099"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977971","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}
Pub Date : 2026-01-13DOI: 10.1016/j.jde.2026.114094
Yang Cai , Changchun Liu , Ming Mei , Zejia Wang
This paper is concerned with the Cauchy problem for 1D compressible Euler equations with spatiotemporal damping in the critical case. We prove the existence of the solutions and their new convergence to the special diffusion waves by the technical time-weighted energy method, where the convergence rates are dependent on the spatial state of the spatiotemporal damping as . These convergence results significantly improve and develop the previous studies of Geng et al. (2020) [10] and Matsumura and Nishihara (2024) [24].
研究具有时空阻尼的一维可压缩欧拉方程在临界情况下的Cauchy问题。我们用技术时间加权能量法证明了该类扩散波解的存在性及其新的收敛性，其中收敛速率依赖于时空阻尼的空间状态为x→±∞。这些收敛结果显著改进和发展了Geng et al.(2020)[10]和Matsumura and Nishihara(2024)[24]的先前研究。
{"title":"Novel convergence of solutions to 1D compressible Euler equations with spatiotemporal damping in critical case","authors":"Yang Cai , Changchun Liu , Ming Mei , Zejia Wang","doi":"10.1016/j.jde.2026.114094","DOIUrl":"10.1016/j.jde.2026.114094","url":null,"abstract":"<div><div>This paper is concerned with the Cauchy problem for 1D compressible Euler equations with spatiotemporal damping in the critical case. We prove the existence of the solutions and their new convergence to the special diffusion waves by the technical time-weighted energy method, where the convergence rates are dependent on the spatial state of the spatiotemporal damping as <span><math><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo></math></span>. These convergence results significantly improve and develop the previous studies of Geng et al. (2020) <span><span>[10]</span></span> and Matsumura and Nishihara (2024) <span><span>[24]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114094"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980539","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}
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