Analysis & PDE最新文献

{"title":"Large-scale regularity for the stationary Navier–Stokes equations over non-Lipschitz boundaries","authors":"Mitsuo Higaki, Christophe Prange, Jinping Zhuge","doi":"10.2140/apde.2024.17.171","DOIUrl":"https://doi.org/10.2140/apde.2024.17.171","url":null,"abstract":"<p>We address the large-scale regularity theory for the stationary Navier–Stokes equations in highly oscillating bumpy John domains. These domains are very rough, possibly with fractals or cusps, at the microscopic scale, but are amenable to the mathematical analysis of the Navier–Stokes equations. We prove a large-scale Calderón–Zygmund estimate, a large-scale Lipschitz estimate, and large-scale higher-order regularity estimates, namely, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>γ</mi></mrow></msup></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>γ</mi></mrow></msup></math> estimates. These nice regularity results are inherited only at mesoscopic scales, and clearly fail in general at the microscopic scales. We emphasize that the large-scale <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>γ</mi></mrow></msup></math> regularity is obtained by using first-order boundary layers constructed via a new argument. The large-scale <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>γ</mi></mrow></msup></math> regularity relies on the construction of second-order boundary layers, which allows for certain boundary data with linear growth at spatial infinity. To the best of our knowledge, our work is the first to carry out such an analysis. In the wake of many works in quantitative homogenization, our results strongly advocate in favor of considering the boundary regularity of the solutions to fluid equations as a multiscale problem, with improved regularity at or above a certain scale. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"14 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752886","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 a family of fully nonlinear integrodifferential operators : from fractional Laplacian to nonlocal Monge–Ampère","authors":"Luis A. Caffarelli, María Soria-Carro","doi":"10.2140/apde.2024.17.243","DOIUrl":"https://doi.org/10.2140/apde.2024.17.243","url":null,"abstract":"<p>We introduce a new family of intermediate operators between the fractional Laplacian and the nonlocal Monge–Ampère introduced by Caffarelli and Silvestre that are given by infimums of integrodifferential operators. Using rearrangement techniques, we obtain representation formulas and give a connection to optimal transport. Finally, we consider a global Poisson problem prescribing data at infinity, and prove existence, uniqueness, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math>-regularity of solutions in the full space. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"38 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752481","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 prescribed curvature problem for entire hypersurfaces in Minkowski space","authors":"Changyu Ren, Zhizhang Wang, Ling Xiao","doi":"10.2140/apde.2024.17.1","DOIUrl":"https://doi.org/10.2140/apde.2024.17.1","url":null,"abstract":"&lt;p&gt;We prove three results in this paper: First, we prove, for a wide class of functions &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;φ&lt;/mi&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi mathvariant=\"double-struck\"&gt;𝕊&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; and &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℝ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℍ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;, there exists a unique, entire, strictly convex, spacelike hypersurface &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; satisfying &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;ψ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; and &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;\u0000&lt;mo&gt;+&lt;/mo&gt;\u0000&lt;mi&gt;φ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∕&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; as &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/math&gt;. Second, when &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;k&lt;/mi&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;, we show the existence and uniqueness of an entire, &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;-convex, spacelike hypersurface &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; satisfying &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;ℳ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;ψ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/ma","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"27 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752593","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":"Shift equivalences through the lens of Cuntz–Krieger algebras","authors":"Toke Meier Carlsen, Adam Dor-On, Søren Eilers","doi":"10.2140/apde.2024.17.345","DOIUrl":"https://doi.org/10.2140/apde.2024.17.345","url":null,"abstract":"<p>Motivated by Williams’ problem of measuring novel differences between shift equivalence (SE) and strong shift equivalence (SSE), we introduce three equivalence relations that provide new ways to obstruct SSE while merely assuming SE. </p><p> Our shift equivalence relations arise from studying graph C*-algebras, where a variety of intermediary equivalence relations naturally arise. As a consequence we realize a goal sought after by Muhly, Pask and Tomforde, measure a delicate difference between SSE and SE in terms of Pimsner dilations for C*-correspondences of adjacency matrices, and use this distinction to refute a proof from a previous paper. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"9 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752473","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 ill-posedness for SQG in critical Sobolev spaces","authors":"In-Jee Jeong, Junha Kim","doi":"10.2140/apde.2024.17.133","DOIUrl":"https://doi.org/10.2140/apde.2024.17.133","url":null,"abstract":"<p>We prove that the inviscid surface quasigeostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">(</mo><msup><mrow><mi mathvariant=\"double-struck\">𝕋</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math> without any solutions in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>∞</mi></mrow></msubsup><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup> </math>. Moreover, we prove strong critical norm inflation for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup></math>-smooth data. Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with the two-dimensional incompressible Euler equations. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"304 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752485","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":"Propagation of singularities for gravity-capillary water waves","authors":"Hui Zhu","doi":"10.2140/apde.2024.17.281","DOIUrl":"https://doi.org/10.2140/apde.2024.17.281","url":null,"abstract":"<p>We obtain two results of propagation for the gravity-capillary water wave system. The first result shows the propagation of oscillations and the spatial decay at infinity; the second result shows a microlocal smoothing effect under the nontrapping condition of the initial free surface. These results extend the works of Craig, Kappeler and Strauss (1995), Wunsch (1999) and Nakamura (2005) to quasilinear dispersive equations. These propagation results are stated for water waves with asymptotically flat free surfaces, of which we also obtain the existence. To prove these results, we generalize the paradifferential calculus of Bony (1979) to weighted Sobolev spaces and develop a semiclassical paradifferential calculus. We also introduce the quasihomogeneous wavefront sets which characterize, in a general manner, the oscillations and the spatial growth/decay of distributions. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"16 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752911","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":"Anisotropic micropolar fluids subject to a uniform microtorque: the stable case","authors":"Antoine Remond-Tiedrez, Ian Tice","doi":"10.2140/apde.2024.17.41","DOIUrl":"https://doi.org/10.2140/apde.2024.17.41","url":null,"abstract":"<p>We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that when the microstructure is inertially oblate (i.e., pancake-like) this equilibrium is nonlinearly asymptotically stable. </p><p> Our proof employs a nonlinear energy method built from the natural energy dissipation structure of the problem. Numerous difficulties arise due to the dissipative-conservative structure of the problem. Indeed, the dissipation fails to be coercive over the energy, which itself is weakly coupled in the sense that, while it provides estimates for the fluid velocity and microstructure angular velocity, it only provides control of two of the six components of the microinertia tensor. To overcome these problems, our method relies on a delicate combination of two distinct tiers of energy-dissipation estimates, together with transport-like advection-rotation estimates for the microinertia. When combined with a quantitative rigidity result for the microinertia, these allow us to deduce the existence of global-in-time decaying solutions near equilibrium. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"17 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752588","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":"Higher rank quantum-classical correspondence","authors":"Joachim Hilgert, Tobias Weich, Lasse L. Wolf","doi":"10.2140/apde.2023.16.2241","DOIUrl":"https://doi.org/10.2140/apde.2023.16.2241","url":null,"abstract":"<p>For a compact Riemannian locally symmetric space <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Γ</mi><mo>∖</mo><mi>G</mi><mo>∕</mo><mi>K</mi></math> of arbitrary rank we determine the location of certain Ruelle–Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle–Taylor resonances and establish a spectral gap which is uniform in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Γ</mi></math> if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi><mo>∕</mo><mi>K</mi></math> is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e., a one-to-one correspondence between horocyclically invariant Ruelle–Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi><mo>∕</mo><mi>K</mi></math>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"10 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683724","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":"Global well-posedness of Vlasov–Poisson-type systems in bounded domains","authors":"Ludovic Cesbron, Mikaela Iacobelli","doi":"10.2140/apde.2023.16.2465","DOIUrl":"https://doi.org/10.2140/apde.2023.16.2465","url":null,"abstract":"<p>In this paper we prove global existence of classical solutions to the Vlasov–Poisson and ionic Vlasov–Poisson models in bounded domains. On the boundary, we consider the specular reflection boundary condition for the Vlasov equation and either homogeneous Dirichlet or Neumann conditions for the Poisson equations. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"34 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683765","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":"Nonexistence of the box dimension for dynamically invariant sets","authors":"Natalia Jurga","doi":"10.2140/apde.2023.16.2385","DOIUrl":"https://doi.org/10.2140/apde.2023.16.2385","url":null,"abstract":"<p>One of the key challenges in the dimension theory of smooth dynamical systems lies in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these three dimensions always coincide. On the other hand, considerable attention has been given to examples of sets invariant under nonconformal dynamics whose Hausdorff and box dimensions do not coincide. These constructions exploit the fact that the Hausdorff and box dimensions quantify size in fundamentally different ways, the former in terms of covers by sets of varying diameters and the latter in terms of covers by sets of fixed diameters. In this article we construct the first example of a dynamically invariant set with distinct lower and upper box dimensions. Heuristically, this says that if size is quantified in terms of covers by sets of equal diameters, a dynamically invariant set can appear bigger when viewed at certain resolutions than at others. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"5 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683763","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}