Analysis & PDE最新文献

{"title":"Variational methods for the kinetic Fokker–Planck equation","authors":"Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat, Matthew Novack","doi":"10.2140/apde.2024.17.1953","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1953","url":null,"abstract":"<p>We develop a functional-analytic approach to the study of the Kramers and kinetic Fokker–Planck equations which parallels the classical <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math> theory of uniformly elliptic equations. In particular, we identify a function space analogous to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math> and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré- and Hörmander-type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup></math> regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker–Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141739030","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}

{"title":"Blow-up of solutions of critical elliptic equations in three dimensions","authors":"Rupert L. Frank, Tobias König, Hynek Kovařík","doi":"10.2140/apde.2024.17.1633","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1633","url":null,"abstract":"<p>We describe the asymptotic behavior of positive solutions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>u</mi></mrow><mrow><mi>𝜀</mi></mrow></msub></math> of the equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>−</mo><mi mathvariant=\"normal\">Δ</mi><mi>u</mi>\u0000<mo>+</mo>\u0000<mi>a</mi><mi>u</mi>\u0000<mo>=</mo> <mn>3</mn><msup><mrow><mi>u</mi></mrow><mrow><mn>5</mn><mo>−</mo><mi>𝜀</mi></mrow></msup></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi>\u0000<mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mn>3</mn></mrow></msup></math> with a homogeneous Dirichlet boundary condition. The function <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>a</mi></math> is assumed to be critical in the sense of Hebey and Vaugon, and the functions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>u</mi></mrow><mrow><mi>𝜀</mi></mrow></msub></math> are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>−</mo><mi mathvariant=\"normal\">Δ</mi><mi>u</mi>\u0000<mo>+</mo>\u0000<mo stretchy=\"false\">(</mo><mi>a</mi>\u0000<mo>+</mo>\u0000<mi>𝜀</mi><mi>V</mi>\u0000<mo stretchy=\"false\">)</mo><mi>u</mi>\u0000<mo>=</mo> <mn>3</mn><msup><mrow><mi>u</mi></mrow><mrow><mn>5</mn></mrow></msup></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi></math>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501261","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}

{"title":"Strong cosmic censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry","authors":"Christoph Kehle, Maxime Van de Moortel","doi":"10.2140/apde.2024.17.1501","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1501","url":null,"abstract":"<p>Motivated by the strong cosmic censorship conjecture in the presence of matter, we study the Einstein equations coupled with a charged/massive scalar field with spherically symmetric characteristic data relaxing to a Reissner–Nordström event horizon. Contrary to the vacuum case, the relaxation rate is conjectured to be <span>slow</span> (nonintegrable), opening the possibility that the matter fields and the metric coefficients <span>blow up in amplitude </span>at the Cauchy horizon, not just in energy. We show that whether this blow-up in amplitude occurs or not depends on a novel <span>oscillation</span>\u0000<span>condition </span>on the event horizon which determines whether or not a resonance is excited dynamically: </p>\u0000<ul>\u0000<li>\u0000<p>If the oscillation condition is satisfied, then the resonance is not excited and we show boundedness and continuous extendibility of the matter fields and the metric across the Cauchy horizon. </p></li>\u0000<li>\u0000<p>If the oscillation condition is violated, then by the <span>combined effect of slow</span>\u0000<span>decay and the resonance being excited</span>, we show that the massive uncharged scalar field blows up in amplitude. </p><p>In a companion paper, we will show that in that case a novel <span>null</span>\u0000<span>contraction singularity </span>forms at the Cauchy horizon, across which the metric is not continuously extendible in the usual sense.</p></li></ul>\u0000<p>Heuristic arguments in the physics literature indicate that the oscillation condition should be satisfied generically on the event horizon. If these heuristics are true, then <span>our result falsifies the</span>\u0000<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math><span>-formulation</span>\u0000<span>of strong cosmic censorship by means of oscillation</span>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501306","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}

{"title":"Connectivity conditions and boundary Poincaré inequalities","authors":"Olli Tapiola, Xavier Tolsa","doi":"10.2140/apde.2024.17.1831","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1831","url":null,"abstract":"<p>Inspired by recent work of Mourgoglou and the second author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi>\u0000<mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup> </math>, with codimension-<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math> Ahlfors–David regular boundaries. First, we prove that if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi></math> satisfies both the local John condition and the exterior corkscrew condition, then <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi></math> also satisfies the Harnack chain condition (and hence is a chord-arc domain). Second, we show that if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi></math> is a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>-sided chord-arc domain, then the boundary <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>∂</mi><mi mathvariant=\"normal\">Ω</mi></math> supports a Heinonen–Koskela-type weak <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math>-Poincaré inequality. We also construct an example of a set <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi>\u0000<mo>⊂</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math> such that the boundary <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>∂</mi><mi mathvariant=\"normal\">Ω</mi></math> is Ahlfors–David regular and supports a weak boundary <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math>-Poincaré inequality but <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi></math> is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141520631","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}

{"title":"Uniform stability in the Euclidean isoperimetric problem for the Allen–Cahn energy","authors":"Francesco Maggi, Daniel Restrepo","doi":"10.2140/apde.2024.17.1761","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1761","url":null,"abstract":"<p>We consider the isoperimetric problem defined on the whole <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> by the Allen–Cahn energy functional. For nondegenerate double-well potentials, we prove sharp quantitative stability inequalities of quadratic type which are uniform in the length scale of the phase transitions. We also derive a rigidity theorem for critical points analogous to the classical Alexandrov theorem for constant mean curvature boundaries. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528896","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}

{"title":"A semiclassical Birkhoff normal form for constant-rank magnetic fields","authors":"Léo Morin","doi":"10.2140/apde.2024.17.1593","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1593","url":null,"abstract":"<p>This paper deals with classical and semiclassical nonvanishing magnetic fields on a Riemannian manifold of arbitrary dimension. We assume that the magnetic field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi>\u0000<mo>=</mo>\u0000<mi>d</mi><mi>A</mi></math> has constant rank and admits a discrete well. On the classical part, we exhibit a harmonic oscillator for the Hamiltonian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi>\u0000<mo>=</mo>\u0000<mo>|</mo><mi>p</mi>\u0000<mo>−</mo>\u0000<mi>A</mi><mo stretchy=\"false\">(</mo><mi>q</mi><mo stretchy=\"false\">)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></math> near the zero-energy surface: the cyclotron motion. On the semiclassical part, we describe the semiexcited spectrum of the magnetic Laplacian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">ℒ</mi></mrow><mrow><mi>ℏ</mi></mrow></msub>\u0000<mo>=</mo> <msup><mrow><mo stretchy=\"false\">(</mo><mi>i</mi><mi>ℏ</mi><mi>d</mi>\u0000<mo>+</mo>\u0000<mi>A</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo>∗</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>i</mi><mi>ℏ</mi><mi>d</mi>\u0000<mo>+</mo>\u0000<mi>A</mi><mo stretchy=\"false\">)</mo></math>. We construct a semiclassical Birkhoff normal form for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">ℒ</mi></mrow><mrow><mi>ℏ</mi></mrow></msub></math> and deduce new asymptotic expansions of the smallest eigenvalues in powers of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℏ</mi></mrow><mrow><mn>1</mn><mo>∕</mo><mn>2</mn></mrow></msup></math> in the limit <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℏ</mi>\u0000<mo>→</mo> <mn>0</mn></math>. In particular we see the influence of the kernel of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> on the spectrum: it raises the energies at order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℏ</mi></mrow><mrow><mn>3</mn><mo>∕</mo><mn>2</mn></mrow></msup></math>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501307","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}

{"title":"A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton","authors":"Benjamin Dodson","doi":"10.2140/apde.2024.17.1693","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1693","url":null,"abstract":"<p>We prove the only blowup solutions to the focusing, quintic nonlinear Schrödinger equation with mass equal to the mass of the soliton are rescaled solitons or the pseudoconformal transformation of those solitons. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501262","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}

{"title":"Noncommutative maximal operators with rough kernels","authors":"Xudong Lai","doi":"10.2140/apde.2024.17.1439","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1439","url":null,"abstract":"<p>This paper is devoted to the study of noncommutative maximal operators with rough kernels. More precisely, we prove the weak-type <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math> boundedness for noncommutative maximal operators with rough kernels. The proof of the weak-type <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math> estimate is based on the noncommutative Calderón–Zygmund decomposition. To deal with the rough kernel, we use the microlocal decomposition in the proofs of both the bad and good functions. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062184","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}

{"title":"On complete embedded translating solitons of the mean curvature flow that are of finite genus","authors":"Graham Smith","doi":"10.2140/apde.2024.17.1175","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1175","url":null,"abstract":"<p>We desingularise the union of three Grim paraboloids along Costa–Hoffman–Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with three ends and arbitrary finite genus. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062177","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}

{"title":"The Landau equation as a gradient Flow","authors":"José A. Carrillo, Matias G. Delgadino, Laurent Desvillettes, Jeremy S.-H. Wu","doi":"10.2140/apde.2024.17.1331","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1331","url":null,"abstract":"<p>We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the energy dissipation inequality (EDI). Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062174","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}