Pub Date : 2026-04-25Epub Date: 2026-01-21DOI: 10.1016/j.jde.2026.114128
Ricardo Alonso , Milana Čolić
In this paper, we study the polyatomic Boltzmann equation based on continuous internal energy, focusing on physically relevant collision kernels of the hard potentials type with integrable angular part. We establish three main results: smoothing effects of the gain collision operator, propagation of velocity and internal energy first-order derivatives of solutions, and exponential decay estimates for singularities of the initial data. These results ultimately lead to a decomposition theorem, showing that any solution splits into a smooth part and a rapidly decaying rough component.
{"title":"Regularity theory for the space homogeneous polyatomic Boltzmann flow","authors":"Ricardo Alonso , Milana Čolić","doi":"10.1016/j.jde.2026.114128","DOIUrl":"10.1016/j.jde.2026.114128","url":null,"abstract":"<div><div>In this paper, we study the polyatomic Boltzmann equation based on continuous internal energy, focusing on physically relevant collision kernels of the hard potentials type with integrable angular part. We establish three main results: smoothing effects of the gain collision operator, propagation of velocity and internal energy first-order derivatives of solutions, and exponential decay estimates for singularities of the initial data. These results ultimately lead to a decomposition theorem, showing that any solution splits into a smooth part and a rapidly decaying rough component.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114128"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036008","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 : 2026-04-25Epub Date: 2026-01-28DOI: 10.1016/j.jde.2026.114142
Jie Chen , Fan Gu , Boling Guo
In this paper, we consider the well-posedness of stochastic S-KdV driven by multiplicative noises in . To get the local well-posedness, we first develop the bilinear and trilinear Bourgain norm estimates of the nonlinear terms with . Then, to overcome regularity problems, we introduce a series of approximation equations with localized nonlinear terms, which are also cutted-off in both the physical and the frequency space. By limitations, these approximation equations will help us get a priori estimate in the Bourgain space and finish the proof of the global well-posedness of the initial system.
{"title":"Stochastic Schrödinger-Korteweg de Vries systems driven by multiplicative noises","authors":"Jie Chen , Fan Gu , Boling Guo","doi":"10.1016/j.jde.2026.114142","DOIUrl":"10.1016/j.jde.2026.114142","url":null,"abstract":"<div><div>In this paper, we consider the well-posedness of stochastic S-KdV driven by multiplicative noises in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>×</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>. To get the local well-posedness, we first develop the bilinear and trilinear Bourgain norm estimates of the nonlinear terms with <span><math><mi>b</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></math></span>. Then, to overcome regularity problems, we introduce a series of approximation equations with localized nonlinear terms, which are also cutted-off in both the physical and the frequency space. By limitations, these approximation equations will help us get a priori estimate in the Bourgain space and finish the proof of the global well-posedness of the initial system.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114142"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074310","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 : 2026-04-25Epub 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":"2026-04-25","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 : 2026-04-25Epub 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":"2026-04-25","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 : 2026-04-25Epub Date: 2026-01-21DOI: 10.1016/j.jde.2026.114132
Zijun Wan , Xiaohua Yao
<div><div>This paper investigates the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-boundedness of wave operators associated with the following nonhomogeneous fourth-order Schrödinger operator on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>:<span><span><span><math><mi>H</mi><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>Δ</mi><mo>.</mo></math></span></span></span> Assuming the real-valued potential <em>V</em> exhibits sufficient decay and regularity, we prove that for all dimensions <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, the wave operators <span><math><msub><mrow><mi>W</mi></mrow><mrow><mo>±</mo></mrow></msub><mo>(</mo><mi>H</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> are bounded on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span>, provided that zero is a regular threshold of <em>H</em>.</div><div>As applications, we derive the sharp <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-<span><math><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup></math></span> dispersive estimates for Schrödinger group <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>i</mi><mi>t</mi><mi>H</mi></mrow></msup></math></span>, as well as for the solutions operators <span><math><mi>cos</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></math></span> and <span><math><mfrac><mrow><mi>sin</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></mrow><mrow><msqrt><mrow><mi>H</mi></mrow></msqrt></mrow></mfrac></math></span> associated with the following beam equations with potentials:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>+</mo><mrow><mo>(</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>Δ</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mo>(</mo><mi>t</mi><mo>,</mo><m
{"title":"The Lp-boundedness of wave operators for nonhomogeneous fourth-order Schrödinger operators in high dimensions","authors":"Zijun Wan , Xiaohua Yao","doi":"10.1016/j.jde.2026.114132","DOIUrl":"10.1016/j.jde.2026.114132","url":null,"abstract":"<div><div>This paper investigates the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-boundedness of wave operators associated with the following nonhomogeneous fourth-order Schrödinger operator on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>:<span><span><span><math><mi>H</mi><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>Δ</mi><mo>.</mo></math></span></span></span> Assuming the real-valued potential <em>V</em> exhibits sufficient decay and regularity, we prove that for all dimensions <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, the wave operators <span><math><msub><mrow><mi>W</mi></mrow><mrow><mo>±</mo></mrow></msub><mo>(</mo><mi>H</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> are bounded on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span>, provided that zero is a regular threshold of <em>H</em>.</div><div>As applications, we derive the sharp <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-<span><math><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup></math></span> dispersive estimates for Schrödinger group <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>i</mi><mi>t</mi><mi>H</mi></mrow></msup></math></span>, as well as for the solutions operators <span><math><mi>cos</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></math></span> and <span><math><mfrac><mrow><mi>sin</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></mrow><mrow><msqrt><mrow><mi>H</mi></mrow></msqrt></mrow></mfrac></math></span> associated with the following beam equations with potentials:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>+</mo><mrow><mo>(</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>Δ</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mo>(</mo><mi>t</mi><mo>,</mo><m","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114132"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036011","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 : 2026-04-25Epub Date: 2026-01-12DOI: 10.1016/j.jde.2026.114097
Bin Qian , Min Wang , Ran Wang , Yimin Xiao
Consider the nonlinear stochastic heat equation where is a Gaussian noise which is white in time and fractional in space with Hurst parameter . The existence and uniqueness of the solutions to this equation were proved by Balan et al. [1] when is an affine function, and by Hu et al. [19] when σ is differentiable with Lipschitz derivative and . In both settings, the Hölder continuity of the solution has been proved by Balan et al. [2] and Hu et al. [19], respectively.
In this paper, we study the asymptotic behavior of the temporal increment for fixed and as , within the framework of [19]. As applications, we derive Khinchin's law of the iterated logarithm, Chung's law of the iterated logarithm, and the quadratic variation of the temporal process , where is fixed.
考虑非线性随机热方程∂u(t,x)∂t=∂2u(t,x)∂x2+σ(u(t,x))W˙（t,x），t>0,x∈R，其中W˙是高斯噪声，在时间上是白的，在空间上是分数的，Hurst参数H∈(14,12)。当σ(u)=au+b是仿射函数时，Balan et al.[1]证明了该方程解的存在唯一性；当σ(0)=0时，σ可与Lipschitz导数微分时，Hu et al.[19]证明了该方程解的存在唯一性。在这两种情况下，分别由Balan et al.[2]和Hu et al.[19]证明了解的Hölder连续性。本文在[19]的框架下，研究了固定t≥0且x∈R为ε↓0时，时间增量u(t+ε,x)−u（t,x）的渐近性。作为应用，我们导出了迭代对数的Khinchin定律，迭代对数的Chung定律，以及时间过程{u(t,x)}t≥0的二次变分，其中x∈R是固定的。
{"title":"Temporal regularity for the nonlinear stochastic heat equation with spatially rough noise","authors":"Bin Qian , Min Wang , Ran Wang , Yimin Xiao","doi":"10.1016/j.jde.2026.114097","DOIUrl":"10.1016/j.jde.2026.114097","url":null,"abstract":"<div><div>Consider the nonlinear stochastic heat equation<span><span><span><math><mfrac><mrow><mo>∂</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>σ</mi><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>)</mo><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is a Gaussian noise which is white in time and fractional in space with Hurst parameter <span><math><mi>H</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>. The existence and uniqueness of the solutions to this equation were proved by Balan et al. <span><span>[1]</span></span> when <span><math><mi>σ</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>u</mi><mo>+</mo><mi>b</mi></math></span> is an affine function, and by Hu et al. <span><span>[19]</span></span> when <em>σ</em> is differentiable with Lipschitz derivative and <span><math><mi>σ</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. In both settings, the Hölder continuity of the solution has been proved by Balan et al. <span><span>[2]</span></span> and Hu et al. <span><span>[19]</span></span>, respectively.</div><div>In this paper, we study the asymptotic behavior of the temporal increment <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>+</mo><mi>ε</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> for fixed <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> as <span><math><mi>ε</mi><mo>↓</mo><mn>0</mn></math></span>, within the framework of <span><span>[19]</span></span>. As applications, we derive Khinchin's law of the iterated logarithm, Chung's law of the iterated logarithm, and the quadratic variation of the temporal process <span><math><msub><mrow><mo>{</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, where <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> is fixed.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114097"},"PeriodicalIF":2.3,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980540","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 : 2026-04-25Epub 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":"2026-04-25","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 : 2026-04-15Epub 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":"2026-04-15","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 : 2026-04-15Epub Date: 2025-12-30DOI: 10.1016/j.jde.2025.114066
Romildo Lima , Liliane Maia , Mayra Soares
We establish the existence of a positive solution to nonlinear Schrödinger equations(PV) for very general potentials V, with positive or zero limit at infinity (allowing convergence from above, below, or oscillations), but imposing no decay rate assumption. Also, the nonlinearities f may satisfy mild hypotheses, including superlinear or asymptotically linear growth at infinity. This is possible due to the application of the classical Monotonicity Trick approach.
{"title":"Advances in solving nonlinear Schrödinger equations with general potentials","authors":"Romildo Lima , Liliane Maia , Mayra Soares","doi":"10.1016/j.jde.2025.114066","DOIUrl":"10.1016/j.jde.2025.114066","url":null,"abstract":"<div><div>We establish the existence of a positive solution to nonlinear Schrödinger equations<span><span><span>(<em>P</em><sub><em>V</em></sub>)</span><span><math><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>u</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mrow><mtext></mtext><mspace></mspace><mtext>in </mtext></mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></math></span></span></span> for very general potentials <em>V</em>, with positive or zero limit at infinity (allowing convergence from above, below, or oscillations), but imposing no decay rate assumption. Also, the nonlinearities <em>f</em> may satisfy mild hypotheses, including superlinear or asymptotically linear growth at infinity. This is possible due to the application of the classical Monotonicity Trick approach.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114066"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882143","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}
In this paper, we investigate global uniform regularity for the compressible Navier-Stokes-Poisson system with damping in the three-dimensional half space under the slip boundary condition. Due to the stabilizing effect induced by the damping term and the construction of a three-layer energy functional framework, we establish global-in-time uniform bounds that are independent of viscosity. These global regularity results and decay rate estimates enable us to rigorously justify the vanishing viscosity limit and derive explicit long-time convergence rates toward the compressible Euler–Poisson system with damping, which serves as a fundamental model describing the motion of compressible, electrostatically interacting fluids in the inviscid regime.
{"title":"Global uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes-Poisson equations with damping under slip boundary condition","authors":"Jincheng Gao , Lianyun Peng , Yuchong Wei , Zheng-an Yao","doi":"10.1016/j.jde.2025.114067","DOIUrl":"10.1016/j.jde.2025.114067","url":null,"abstract":"<div><div>In this paper, we investigate global uniform regularity for the compressible Navier-Stokes-Poisson system with damping in the three-dimensional half space under the slip boundary condition. Due to the stabilizing effect induced by the damping term and the construction of a three-layer energy functional framework, we establish global-in-time uniform bounds that are independent of viscosity. These global regularity results and decay rate estimates enable us to rigorously justify the vanishing viscosity limit and derive explicit long-time convergence rates toward the compressible Euler–Poisson system with damping, which serves as a fundamental model describing the motion of compressible, electrostatically interacting fluids in the inviscid regime.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114067"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882141","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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