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Full asymptotic expansion of the permeability matrix of a dilute periodic porous medium
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.029
F. Feppon
{"title":"Full asymptotic expansion of the permeability matrix of a dilute periodic porous medium","authors":"F. Feppon","doi":"10.1016/j.jde.2024.11.029","DOIUrl":"10.1016/j.jde.2024.11.029","url":null,"abstract":"<div><div>We compute full asymptotic expansions of the permeability matrix of a laminar fluid flowing through a periodic array of small solid particles. The derivation considers obstacles with arbitrary shape in arbitrary space dimension. In the first step, we use hydrodynamics layer potential theory to obtain the asymptotic expansion of the velocity and pressure fields across the periodic array. The terms of these expansions can be computed through a procedure involving a cascade of exterior and interior problems. In the second step, we deduce the asymptotic expansion of the permeability matrix. The derivation requires evaluating Hadamard finite part integrals and tensors depending on the values of the fundamental solution or its derivatives on the faces of the unit cell. We verify that our expansions agree to the leading order with the expressions found by Hasimoto <span><span>[24]</span></span> in the case of spherical obstacles in two and three dimensions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 178-237"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745975","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
Neumann problem for fractional Ginzburg-Landau equation on a upper- right quarter plane
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.033
J.F. Carreño-Diaz, E.I. Kaikina
{"title":"Neumann problem for fractional Ginzburg-Landau equation on a upper- right quarter plane","authors":"J.F. Carreño-Diaz,&nbsp;E.I. Kaikina","doi":"10.1016/j.jde.2024.11.033","DOIUrl":"10.1016/j.jde.2024.11.033","url":null,"abstract":"<div><div>We consider the initial-boundary value problem for the Ginzburg-Landau equation with fractional Laplacian on a upper-right quarter plane.</div><div>We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the Sobolev spaces to the case of a Neumann type of boundary data. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 258-304"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745977","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
Symbolic dynamics of planar piecewise smooth vector fields
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.031
Tiago Carvalho , André do Amaral Antunes
{"title":"Symbolic dynamics of planar piecewise smooth vector fields","authors":"Tiago Carvalho ,&nbsp;André do Amaral Antunes","doi":"10.1016/j.jde.2024.11.031","DOIUrl":"10.1016/j.jde.2024.11.031","url":null,"abstract":"<div><div>It is well known that many results obtained for piecewise smooth vector fields do not have an analogous for smooth vector fields and vice-versa. These differences are generated by the non-uniqueness of trajectory passing through a point. Inspired by the classical fact that one-dimensional discrete dynamic systems can produce chaotic behavior, we construct a conjugation between shift maps and piecewise smooth vector fields presenting homoclinic loops which are associated to symbols in such a way that the flow restricted to a homoclinic loop is codified with a symbol. The construction of the topological conjugation between the quoted piecewise smooth vector fields and the respective shift spaces needs several technicality which were solved considering a specific family of piecewise smooth vector fields (<span><span>Theorem A</span></span>) and then generalizing the result for an entire class of piecewise smooth vector fields (<span><span>Theorem B</span></span>). By means of the results obtained and the techniques employed, a new perspective on the study of piecewise smooth vector fields is brought to light and, through already established results for discrete dynamic systems, we will be able to obtain results regarding piecewise smooth vector fields.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 150-174"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746149","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
Small data non-linear wave equation numerology: The role of asymptotics
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.021
Istvan Kadar
{"title":"Small data non-linear wave equation numerology: The role of asymptotics","authors":"Istvan Kadar","doi":"10.1016/j.jde.2024.11.021","DOIUrl":"10.1016/j.jde.2024.11.021","url":null,"abstract":"<div><div>Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not <em>critical</em> according to our classification. Furthermore, we gave two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in <span><span>[13]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 305-373"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745978","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
Uniform regularity for incompressible MHD equations in a bounded domain with curved boundary in 3D
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.028
Yingzhi Du, Tao Luo
{"title":"Uniform regularity for incompressible MHD equations in a bounded domain with curved boundary in 3D","authors":"Yingzhi Du,&nbsp;Tao Luo","doi":"10.1016/j.jde.2024.11.028","DOIUrl":"10.1016/j.jde.2024.11.028","url":null,"abstract":"<div><div>For the initial boundary problem of the incompressible MHD equations in a bounded domain with general curved boundary in 3D with the general Navier-slip boundary conditions for the velocity field and the perfect conducting condition for the magnetic field, we establish the uniform regularity of conormal Sobolev norms and Lipschitz norms to addressing the anisotropic regularity of tangential and normal directions, which enable us to prove the vanishing dissipation limit as the viscosity and the magnetic diffusion coefficients tend to zero. We overcome the difficulties caused by the intricate interaction of boundary curvature, velocity field and magnetic fields and resolve the issue caused by the problem that the viscosity and the magnetic diffusion coefficients are not required to equal.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 175-252"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746150","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
Higher Hölder regularity for a subquadratic nonlocal parabolic equation
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.024
Prashanta Garain , Erik Lindgren , Alireza Tavakoli
{"title":"Higher Hölder regularity for a subquadratic nonlocal parabolic equation","authors":"Prashanta Garain ,&nbsp;Erik Lindgren ,&nbsp;Alireza Tavakoli","doi":"10.1016/j.jde.2024.11.024","DOIUrl":"10.1016/j.jde.2024.11.024","url":null,"abstract":"<div><div>In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>.</mo></math></span></span></span> Here, <span><math><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> is the fractional <em>p</em>-Laplacian, <span><math><mn>0</mn><mo>&lt;</mo><mi>s</mi><mo>&lt;</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mn>2</mn></math></span>. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 253-290"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746151","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
Non-uniqueness of admissible weak solutions to the two-dimensional pressureless Euler system
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.032
Feimin Huang , Jiajin Shi , Yi Wang
{"title":"Non-uniqueness of admissible weak solutions to the two-dimensional pressureless Euler system","authors":"Feimin Huang ,&nbsp;Jiajin Shi ,&nbsp;Yi Wang","doi":"10.1016/j.jde.2024.11.032","DOIUrl":"10.1016/j.jde.2024.11.032","url":null,"abstract":"<div><div>We study Riemann problem for the two-dimensional (2D) pressureless Euler system with planar Riemann initial data. It is proved that there exist infinitely many bounded admissible weak solutions to the 2D Riemann problem by the method of convex integration. Meanwhile, the corresponding one-dimensional (1D) Riemann problem admits a unique measure-valued solution (so-called <em>δ</em>-shock) under the Oleĭnik's entropy condition and an additional energy condition, which implies the non-existence of 1D bounded admissible weak solutions with energy condition (cf. <span><span>[19]</span></span>).</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 238-257"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745976","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
An integrable pseudospherical equation with pseudo-peakon solutions
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-29 DOI: 10.1016/j.jde.2024.11.030
Priscila Leal da Silva , Igor Leite Freire , Nazime Sales Filho
{"title":"An integrable pseudospherical equation with pseudo-peakon solutions","authors":"Priscila Leal da Silva ,&nbsp;Igor Leite Freire ,&nbsp;Nazime Sales Filho","doi":"10.1016/j.jde.2024.11.030","DOIUrl":"10.1016/j.jde.2024.11.030","url":null,"abstract":"<div><div>We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From the symmetries, we obtain invariant solutions that provide explicit metrics for the surfaces. These solutions are unbounded and often appear in mirrored pairs. We introduce the “collage” method, which uses conserved quantities to remove unbounded parts and smoothly join the solutions, leading to weak solutions consistent with the conserved quantities. As a result we get pseudo-peakons, which are smoother than Camassa-Holm peakons. Additionally, we apply a Miura-type transformation to relate our equation to the Degasperis-Procesi equation, allowing us to recover peakon and shock-peakon solutions for it from the solutions of the other equation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 291-323"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746152","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 splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D 关于三维粘性和非阻力不可压缩磁流体动力学方程自由边界问题的飞溅奇点
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-28 DOI: 10.1016/j.jde.2024.11.026
Guangyi Hong , Tao Luo , Zhonghao Zhao
{"title":"On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D","authors":"Guangyi Hong ,&nbsp;Tao Luo ,&nbsp;Zhonghao Zhao","doi":"10.1016/j.jde.2024.11.026","DOIUrl":"10.1016/j.jde.2024.11.026","url":null,"abstract":"<div><div>In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in <span><span>[14, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2019]</span></span> from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 40-80"},"PeriodicalIF":2.4,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722411","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
Global analyticity and the lower bounds of analytic radius for the Chaplygin gas equations with source terms 带有源项的查普利金气体方程的全局解析性和解析半径下限
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2024-11-28 DOI: 10.1016/j.jde.2024.11.027
Zhengyan Liu , Xinglong Wu , Boling Guo
{"title":"Global analyticity and the lower bounds of analytic radius for the Chaplygin gas equations with source terms","authors":"Zhengyan Liu ,&nbsp;Xinglong Wu ,&nbsp;Boling Guo","doi":"10.1016/j.jde.2024.11.027","DOIUrl":"10.1016/j.jde.2024.11.027","url":null,"abstract":"<div><div>This paper is devoted to studying the global existence and the analytic radius of analytic solutions to the Chaplygin gas equations with source terms. If the initial data belongs to Gevrey spaces and it is sufficiently small, we show the solution has the global persistent property in Gevrey spaces. In particular, we obtain uniform lower bounds on the spatial analytic radius which is given by <span><math><mi>C</mi><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>C</mi><mi>t</mi></mrow></msup></math></span>, for some constant <span><math><mi>C</mi><mo>&gt;</mo><mn>0</mn></math></span>, this tells us that the decay rate of the analytic radius is at most a single exponential decay, which is the slowest decay rate of lower bounds on the analytic radius compared with the double and triple exponential decay of analytic radius derived by Levermore, Bardos, et al. (see <span><span>Remark 1.2</span></span>). Our method is based on the Fourier transformation and Gevrey-class regularity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 81-113"},"PeriodicalIF":2.4,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722412","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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