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Stability analysis of the incompressible porous media equation and the Stokes transport system via energy structure. 不可压缩多孔介质方程及Stokes输运系统的能量结构稳定性分析。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-05-28 DOI: 10.1007/s00526-025-03029-y
Jaemin Park
{"title":"Stability analysis of the incompressible porous media equation and the Stokes transport system via energy structure.","authors":"Jaemin Park","doi":"10.1007/s00526-025-03029-y","DOIUrl":"https://doi.org/10.1007/s00526-025-03029-y","url":null,"abstract":"<p><p>In this paper, we revisit asymptotic stability for the two-dimensional incompressible porous media equation and the Stokes transport system in a periodic channel. It is well-known that a stratified density, which strictly decreases in the vertical direction, is asymptotically stable under sufficiently small and smooth perturbations. We provide improvements in the regularity assumptions on the perturbation and in the convergence rate. Unlike the standard approach for stability analysis relying on linearized equations, we directly address the nonlinear problem by exploiting the energy structure of each system. While it is widely known that the potential energy is a Lyapunov functional in both systems, our key observation is that the second derivative of the potential energy reveals a (degenerate) coercive structure, which arises from the fact that the solution converges to the minimizer of the energy.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 5","pages":"169"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12119689/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144198266","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}
引用次数: 0
Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities. 非局部波动方程的最优Runge近似和多齐次非线性的唯一确定。
IF 2 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-10-10 DOI: 10.1007/s00526-025-03124-0
Yi-Hsuan Lin, Teemu Tyni, Philipp Zimmermann
{"title":"Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities.","authors":"Yi-Hsuan Lin, Teemu Tyni, Philipp Zimmermann","doi":"10.1007/s00526-025-03124-0","DOIUrl":"10.1007/s00526-025-03124-0","url":null,"abstract":"<p><p>The main purpose of this article is to establish the Runge-type approximation in <math> <mrow><msup><mi>L</mi> <mn>2</mn></msup> <mrow><mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup><mover><mi>H</mi> <mo>~</mo></mover> <mi>s</mi></msup> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> </mrow> </math> for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calderón problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calderón problem of nonlocal wave equations with polyhomogeneous nonlinearities.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 9","pages":"280"},"PeriodicalIF":2.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12513988/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145279017","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}
引用次数: 0
The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices 通过基本矩阵的特征值求欧几里得球中最小 Legendrian 子满足的刚性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-19 DOI: 10.1007/s00526-024-02822-5
Pei-Yi Wu, Ling Yang
{"title":"The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices","authors":"Pei-Yi Wu, Ling Yang","doi":"10.1007/s00526-024-02822-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02822-5","url":null,"abstract":"<p>In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu’s inequality (Lu in J Funct Anal 261:1284–1308, 2011) on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"37 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250268","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}
引用次数: 0
Isoperimetry and the properness of weak inverse mean curvature flow 等压法和弱反平均曲率流的适当性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-18 DOI: 10.1007/s00526-024-02832-3
Kai Xu
{"title":"Isoperimetry and the properness of weak inverse mean curvature flow","authors":"Kai Xu","doi":"10.1007/s00526-024-02832-3","DOIUrl":"https://doi.org/10.1007/s00526-024-02832-3","url":null,"abstract":"<p>We prove a new existence theorem for proper solutions of Huisken and Ilmanen’s weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250269","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}
引用次数: 0
On the $$L^{p}$$ dual Minkowski problem for $$-1<0$$ 关于$$-1<0$$的$$L^{p}$$对偶闵科夫斯基问题
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-17 DOI: 10.1007/s00526-024-02806-5
Stephanie Mui
{"title":"On the $$L^{p}$$ dual Minkowski problem for $$-1<0$$","authors":"Stephanie Mui","doi":"10.1007/s00526-024-02806-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02806-5","url":null,"abstract":"<p>The <span>(L^{p})</span> dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the <span>(L^{p})</span> dual Minkowski problem, asks about existence of a convex body with prescribed <span>(L^{p})</span> dual curvature measure. This question unifies the previously disjoint <span>(L^{p})</span> Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the <span>(L^{p})</span> dual Minkowski problem for the case of <span>(q&lt;p+1)</span>, <span>(-1&lt;p&lt;0)</span>, and <span>(pne q)</span> for even measures.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"11 14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250272","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}
引用次数: 0
Hele-Shaw flow as a singular limit of a Keller-Segel system with nonlinear diffusion 作为具有非线性扩散的凯勒-西格尔系统奇异极限的赫勒-肖流
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-17 DOI: 10.1007/s00526-024-02826-1
Antoine Mellet
{"title":"Hele-Shaw flow as a singular limit of a Keller-Segel system with nonlinear diffusion","authors":"Antoine Mellet","doi":"10.1007/s00526-024-02826-1","DOIUrl":"https://doi.org/10.1007/s00526-024-02826-1","url":null,"abstract":"<p>We study a singular limit of the classical parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The main result is the <span>(Gamma )</span> convergence of the corresponding energy functional toward the perimeter functional. Following recent work on this topic, we then prove that under an energy convergence assumption, the solution of the PKS model converges to a solution of the Hele-Shaw free boundary problem with surface tension, which describes the evolution of the interface separating regions with high density from those with low density. This result complements a recent work by the author with I. Kim and Y. Wu, in which the same free boundary problem is derived from the congested PKS model (which includes a density constraint <span>(rho le 1)</span> and a pressure term): It shows that the congestion constraint is not necessary to observe phase separation and surface tension phenomena.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250270","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}
引用次数: 0
Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms 带有加权非局部源和梯度吸收项的准线性椭圆微分不等式的利乌维尔类型定理
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-16 DOI: 10.1007/s00526-024-02821-6
Ye Du, Zhong Bo Fang
{"title":"Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms","authors":"Ye Du, Zhong Bo Fang","doi":"10.1007/s00526-024-02821-6","DOIUrl":"https://doi.org/10.1007/s00526-024-02821-6","url":null,"abstract":"<p>This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a strongly <i>p</i>-coercive elliptic differential inequality with weighted nonlocal source and gradient absorption terms in the whole space. Under the condition that the positive weight in the absorption term is either a sufficiently small constant or more general, we establish new Liouville type results containing the critical case. The key ingredient in the proof is the rescaled test function method developed by Mitidieri and Pohozaev.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250274","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}
引用次数: 0
Convergence of semi-convex functions in CAT(1)-spaces CAT(1)-spaces 中半凸函数的收敛性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-16 DOI: 10.1007/s00526-024-02823-4
Hedvig Gál, Miklós Pálfia
{"title":"Convergence of semi-convex functions in CAT(1)-spaces","authors":"Hedvig Gál, Miklós Pálfia","doi":"10.1007/s00526-024-02823-4","DOIUrl":"https://doi.org/10.1007/s00526-024-02823-4","url":null,"abstract":"<p>We generalize the results of Kuwae–Shioya and Bačák on Mosco convergence established for CAT(0)-spaces to the CAT(1)-setting, so that Mosco convergence implies convergence of resolvents which in turn imply convergence of gradient flows for lower-semicontinuous semi-convex functions. Our techniques utilize weak convergence in CAT(1)-spaces and also cover asymptotic relations of sequences of such spaces introduced by Kuwae-Shioya, including Gromov–Hausdorff limits.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"51 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250273","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}
引用次数: 0
The anisotropic Gaussian isoperimetric inequality and Ehrhard symmetrization 各向异性高斯等周不等式和艾哈德对称性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-09 DOI: 10.1007/s00526-024-02818-1
Kuan-Ting Yeh
{"title":"The anisotropic Gaussian isoperimetric inequality and Ehrhard symmetrization","authors":"Kuan-Ting Yeh","doi":"10.1007/s00526-024-02818-1","DOIUrl":"https://doi.org/10.1007/s00526-024-02818-1","url":null,"abstract":"<p>In this paper, we prove the isoperimetric inequality for the anisotropic Gaussian measure and characterize the cases of equality. We also find an example that shows Ehrhard symmetrization fails to decrease for the anisotropic Gaussian perimeter and gives a new inequality that includes an error term. This new inequality, in particular, gives us a hint to prove a uniqueness result for the anisotropic Ehrhard symmetrization.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192550","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}
引用次数: 0
Nonlinear scalar field $$(p_{1}, p_{2})$$ -Laplacian equations in $$mathbb {R}^{N}$$ : existence and multiplicity $$mathbb{R}^{N}$$中的非线性标量场$$(p_{1}, p_{2})$$ -拉普拉斯方程：存在性与多重性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-02 DOI: 10.1007/s00526-024-02797-3
Vincenzo Ambrosio
{"title":"Nonlinear scalar field $$(p_{1}, p_{2})$$ -Laplacian equations in $$mathbb {R}^{N}$$ : existence and multiplicity","authors":"Vincenzo Ambrosio","doi":"10.1007/s00526-024-02797-3","DOIUrl":"https://doi.org/10.1007/s00526-024-02797-3","url":null,"abstract":"<p>In this paper, we deal with the following class of <span>((p_{1}, p_{2}))</span>-Laplacian problems: </p><span>$$begin{aligned} left{ begin{array}{ll} -Delta _{p_{1}}u-Delta _{p_{2}}u= g(u) text{ in } mathbb {R}^{N}, uin W^{1, p_{1}}(mathbb {R}^{N})cap W^{1, p_{2}}(mathbb {R}^{N}), end{array} right. end{aligned}$$</span><p>where <span>(Nge 2)</span>, <span>(1&lt;p_{1}&lt;p_{2}le N)</span>, <span>(Delta _{p_{i}})</span> is the <span>(p_{i})</span>-Laplacian operator, for <span>(i=1, 2)</span>, and <span>(g:mathbb {R}rightarrow mathbb {R})</span> is a Berestycki-Lions type nonlinearity. Using appropriate variational arguments, we obtain the existence of a ground state solution. In particular, we provide three different approaches to deduce this result. Finally, we prove the existence of infinitely many radially symmetric solutions. Our results improve and complement those that have appeared in the literature for this class of problems. Furthermore, the arguments performed throughout the paper are rather flexible and can be also applied to study other <i>p</i>-Laplacian and <span>((p_1, p_2))</span>-Laplacian equations with general nonlinearities.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"49 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192551","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}
引用次数: 0
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