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Semiclassical eigenvalue estimates under magnetic steps 磁场阶跃下的半经典特征值估计
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.535
Wafaa Assaad, Bernard Helffer, Ayman Kachmar
{"title":"Semiclassical eigenvalue estimates under magnetic steps","authors":"Wafaa Assaad, Bernard Helffer, Ayman Kachmar","doi":"10.2140/apde.2024.17.535","DOIUrl":"https://doi.org/10.2140/apde.2024.17.535","url":null,"abstract":"<p>We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues for the Dirichlet magnetic Laplacian with a nonuniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit, which also corresponds to a large magnetic field limit and is valid under a geometric assumption on the curvature of the discontinuity curve. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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 a spatially inhomogeneous nonlinear Fokker–Planck equation : Cauchy problem and diffusion asymptotics 关于空间非均质非线性福克-普朗克方程：考奇问题和扩散渐近学
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.379
Francesca Anceschi, Yuzhe Zhu
{"title":"On a spatially inhomogeneous nonlinear Fokker–Planck equation : Cauchy problem and diffusion asymptotics","authors":"Francesca Anceschi, Yuzhe Zhu","doi":"10.2140/apde.2024.17.379","DOIUrl":"https://doi.org/10.2140/apde.2024.17.379","url":null,"abstract":"<p>We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker–Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global-in-time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy, and barrier function methods. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"22 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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 L∞ estimates for Monge–Ampère and Hessian equations on nef classes 论 NEF 类上 Monge-Ampère 和 Hessian 方程的 L∞ 估计值
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.749
Bin Guo, Duong H. Phong, Freid Tong, Chuwen Wang
{"title":"On L∞ estimates for Monge–Ampère and Hessian equations on nef classes","authors":"Bin Guo, Duong H. Phong, Freid Tong, Chuwen Wang","doi":"10.2140/apde.2024.17.749","DOIUrl":"https://doi.org/10.2140/apde.2024.17.749","url":null,"abstract":"<p>The PDE approach developed earlier by the first three authors for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math> estimates for fully nonlinear equations on Kähler manifolds is shown to apply as well to Monge–Ampère and Hessian equations on nef classes. In particular, one obtains a new proof of the estimates of Boucksom, Eyssidieux, Guedj and Zeriahi (2010) and Fu, Guo and Song (2020) for the Monge–Ampère equation, together with their generalization to Hessian equations. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"66 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential 最小上梯度的曲线特征和索波列夫微分的构造
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.455
Sylvester Eriksson-Bique, Elefterios Soultanis
{"title":"Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential","authors":"Sylvester Eriksson-Bique, Elefterios Soultanis","doi":"10.2140/apde.2024.17.455","DOIUrl":"https://doi.org/10.2140/apde.2024.17.455","url":null,"abstract":"<p>We represent minimal upper gradients of Newtonian functions, in the range <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn>\u0000<mo>≤</mo>\u0000<mi>p</mi>\u0000<mo>&lt;</mo>\u0000<mi>∞</mi></math>, by maximal directional derivatives along “generic” curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. </p><p> The arising <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger’s structure in the presence of a Poincaré inequality. In particular, it exists whenever the space is metrically doubling. It is moreover compatible with, and gives a geometric interpretation of, Gigli’s abstract differentiable structure, whenever it exists. The <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-weak charts give rise to a finite-dimensional <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"51 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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 blowup for the supercritical quadratic wave equation 关于超临界二次波方程的炸毁问题
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.617
Elek Csobo, Irfan Glogić, Birgit Schörkhuber
{"title":"On blowup for the supercritical quadratic wave equation","authors":"Elek Csobo, Irfan Glogić, Birgit Schörkhuber","doi":"10.2140/apde.2024.17.617","DOIUrl":"https://doi.org/10.2140/apde.2024.17.617","url":null,"abstract":"<p>We study singularity formation for the quadratic wave equation in the energy supercritical case, i.e., for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≥</mo> <mn>7</mn></math>. We find in closed form a new, nontrivial, radial, self-similar blow-up solution <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>u</mi></mrow><mrow><mo>∗</mo></mrow></msup></math> which exists for all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≥</mo> <mn>7</mn></math>. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>=</mo> <mn>9</mn></math>, we study the stability of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>u</mi></mrow><mrow><mo>∗</mo></mrow></msup></math> without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>u</mi></mrow><mrow><mo>∗</mo></mrow></msup> </math>. In similarity coordinates, this family represents a codimension-1 Lipschitz manifold modulo translation symmetries. The stability analysis relies on delicate spectral analysis for a non-self-adjoint operator. In addition, in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>=</mo> <mn>7</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>=</mo> <mn>9</mn></math>, we prove nonradial stability of the well-known ODE blow-up solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"21 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Necessary density conditions for sampling and interpolation in spectral subspaces of elliptic differential operators 椭圆微分算子谱子空间中采样和插值的必要密度条件
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.587
Karlheinz Gröchenig, Andreas Klotz
{"title":"Necessary density conditions for sampling and interpolation in spectral subspaces of elliptic differential operators","authors":"Karlheinz Gröchenig, Andreas Klotz","doi":"10.2140/apde.2024.17.587","DOIUrl":"https://doi.org/10.2140/apde.2024.17.587","url":null,"abstract":"<p>We prove necessary density conditions for sampling in spectral subspaces of a second-order uniformly elliptic differential operator on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>d</mi></mrow></msup></math> with slowly oscillating symbol. For constant-coefficient operators, these are precisely Landau’s necessary density conditions for bandlimited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even exists. Our results prove the existence of a suitable critical sampling density and compute it in terms of the geometry defined by the elliptic operator. In dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>=</mo> <mn>1</mn></math>, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and we obtain a new critical density for variable bandwidth. The methods are a combination of the spectral theory and the regularity theory of elliptic partial differential operators, some elements of limit operators, certain compactifications of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>d</mi></mrow></msup> </math>, and the theory of reproducing kernel Hilbert spaces. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"21 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Explicit formula of radiation fields of free waves with applications on channel of energy 自由波辐射场的显式公式及其在能量通道中的应用
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.723
Liang Li, Ruipeng Shen, Lijuan Wei
{"title":"Explicit formula of radiation fields of free waves with applications on channel of energy","authors":"Liang Li, Ruipeng Shen, Lijuan Wei","doi":"10.2140/apde.2024.17.723","DOIUrl":"https://doi.org/10.2140/apde.2024.17.723","url":null,"abstract":"<p>We give a few explicit formulas regarding the radiation fields of linear free waves. We then apply these formulas on the channel-of-energy theory. We characterize all the radial weakly nonradiative solutions in all dimensions and give a few new exterior energy estimates. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"66 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Strichartz inequalities with white noise potential on compact surfaces 紧凑表面上具有白噪声势的斯特里哈茨不等式
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.421
Antoine Mouzard, Immanuel Zachhuber
{"title":"Strichartz inequalities with white noise potential on compact surfaces","authors":"Antoine Mouzard, Immanuel Zachhuber","doi":"10.2140/apde.2024.17.421","DOIUrl":"https://doi.org/10.2140/apde.2024.17.421","url":null,"abstract":"<p>We prove Strichartz inequalities for the Schrödinger equation and the wave equation with multiplicative noise on a two-dimensional manifold. This relies on the Anderson Hamiltonian described using high-order paracontrolled calculus. As an application, it gives a low-regularity solution theory for the associated nonlinear equations. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"51 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Smooth extensions for inertial manifolds of semilinear parabolic equations 半线性抛物方程惯性流形的光滑扩展
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.499
Anna Kostianko, Sergey Zelik
{"title":"Smooth extensions for inertial manifolds of semilinear parabolic equations","authors":"Anna Kostianko, Sergey Zelik","doi":"10.2140/apde.2024.17.499","DOIUrl":"https://doi.org/10.2140/apde.2024.17.499","url":null,"abstract":"<p>The paper is devoted to a comprehensive study of smoothness of inertial manifolds (IMs) for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>𝜀</mi></mrow></msup></math>-regularity for such manifolds (for some positive, but small <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi></math>). Nevertheless, as shown in the paper, under natural assumptions, the obstacles to the existence of a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math>-smooth inertial manifold (where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>∈</mo>\u0000<mi>ℕ</mi></math> is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>𝜀</mi></mrow></msup></math>-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"18 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Arnold’s variational principle and its application to the stability of planar vortices 阿诺德变分原理及其在平面涡旋稳定性中的应用
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2024-03-06 DOI: 10.2140/apde.2024.17.681
Thierry Gallay, Vladimír Šverák
{"title":"Arnold’s variational principle and its application to the stability of planar vortices","authors":"Thierry Gallay, Vladimír Šverák","doi":"10.2140/apde.2024.17.681","DOIUrl":"https://doi.org/10.2140/apde.2024.17.681","url":null,"abstract":"<p>We consider variational principles related to V. I. Arnold’s stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the Navier–Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen’s vortex, providing a new stability proof with stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"21 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140056045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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