Archive for Rational Mechanics and Analysis最新文献_第6页

Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System 凯勒-西格尔系统中无限时间炸裂的存在性和稳定性

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-06-18 DOI: 10.1007/s00205-024-02006-7

Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei

{"title":"Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System","authors":"Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei","doi":"10.1007/s00205-024-02006-7","DOIUrl":"10.1007/s00205-024-02006-7","url":null,"abstract":"<div><p>Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system </p><div><figure><div><div><picture><img></picture></div></div></figure></div><p> We consider the critical mass case <span>(int _{{mathbb {R}}^2} u_0(x), textrm{d}x = 8pi )</span>, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function <span>(u_0^*)</span> with mass <span>(8pi )</span> such that for any initial condition <span>(u_0)</span> sufficiently close to <span>(u_0^*)</span> and mass <span>(8pi )</span>, the solution <i>u</i>(<i>x</i>, <i>t</i>) of (<span>(*)</span>) is globally defined and blows-up in infinite time. As <span>(trightarrow +infty )</span> it has the approximate profile </p><div><div><span>$$begin{aligned} u(x,t) approx frac{1}{lambda ^2(t)} Uleft( frac{x-xi (t)}{lambda (t)} right) , quad U(y)= frac{8}{(1+|y|^2)^2}, end{aligned}$$</span></div></div><p>where <span>(lambda (t) approx frac{c}{sqrt{log t}})</span>, <span>(xi (t)rightarrow q)</span> for some <span>(c>0)</span> and <span>(qin {mathbb {R}}^2)</span>. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02006-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Regularity for Nonuniformly Elliptic Equations with (p,!q)-Growth and Explicit (x,!u)-Dependence 具有$$p,!q$$增长和明确$$x,!u$$依赖性的非均匀椭圆方程的正则性

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-06-17 DOI: 10.1007/s00205-024-01982-0

Giovanni Cupini, Paolo Marcellini, Elvira Mascolo

引用次数: 0

Conditional (L^{infty }) Estimates for the Non-cutoff Boltzmann Equation in a Bounded Domain 有界域中非截止波尔兹曼方程的条件 $$L^{infty }$$ 估计值

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-06-06 DOI: 10.1007/s00205-024-02002-x

Zhimeng Ouyang, Luis Silvestre

引用次数: 0

Correction to: Stable Singularity Formation for the Keller-Segel System in Three Dimensions 更正：三维凯勒-西格尔系统的稳定奇点形成

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-06-05 DOI: 10.1007/s00205-024-02004-9

Irfan Glogić, Birgit Schörkhuber

引用次数: 0

Correction to: Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation 更正：修正科特维格-德-弗里斯方程自相似解在膨胀时的扰动

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-06-05 DOI: 10.1007/s00205-024-02005-8

Simão Correia, Raphaël Côte

引用次数: 0

Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem 二维可压缩等离子体-真空界面问题的好拟性

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-06-04 DOI: 10.1007/s00205-024-02001-y

Alessandro Morando, Paolo Secchi, Yuri Trakhinin, Paola Trebeschi, Difan Yuan

{"title":"Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem","authors":"Alessandro Morando, Paolo Secchi, Yuri Trakhinin, Paola Trebeschi, Difan Yuan","doi":"10.1007/s00205-024-02001-y","DOIUrl":"10.1007/s00205-024-02001-y","url":null,"abstract":"<div><p>We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02001-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Gradient Blow-Up for Dispersive and Dissipative Perturbations of the Burgers Equation 布尔格斯方程的分散扰动和耗散扰动的梯度放大

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-05-30 DOI: 10.1007/s00205-024-01985-x

Sung-Jin Oh, Federico Pasqualotto

{"title":"Gradient Blow-Up for Dispersive and Dissipative Perturbations of the Burgers Equation","authors":"Sung-Jin Oh, Federico Pasqualotto","doi":"10.1007/s00205-024-01985-x","DOIUrl":"10.1007/s00205-024-01985-x","url":null,"abstract":"<div><p>We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order <span>(alpha )</span>, and the fractal Burgers equation of order <span>(beta )</span>, where <span>(alpha , beta in [0,1))</span>, and the Whitham equation. For all <span>(alpha , beta in [0,1))</span>, we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a “shock-like” singularity. Moreover, we provide an asymptotic description of the blow-up. To the best of our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range <span>(alpha in [2/3, 1))</span>, as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range <span>(beta in [2/3, 1))</span>. Our construction is based on modulation theory, where the well-known smooth self-similar solutions to the inviscid Burgers equation are used as profiles. A somewhat amusing point is that the profiles that are less stable under initial data perturbations (in that the number of unstable directions is larger) are more stable under perturbations of the equation (in that higher order dispersive and/or dissipative terms are allowed) due to their slower rates of concentration. Another innovation of this article, which may be of independent interest, is the development of a streamlined weighted <span>(L^{2})</span>-based approach (in lieu of the characteristic method) for establishing the sharp spatial behavior of the solution in self-similar variables, which leads to the sharp Hölder regularity of the solution up to the blow-up time.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195264","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

The Feynman–Lagerstrom Criterion for Boundary Layers 边界层的费曼-拉格斯特罗姆准则

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-05-30 DOI: 10.1007/s00205-024-01991-z

Theodore D. Drivas, Sameer Iyer, Trinh T. Nguyen

{"title":"The Feynman–Lagerstrom Criterion for Boundary Layers","authors":"Theodore D. Drivas, Sameer Iyer, Trinh T. Nguyen","doi":"10.1007/s00205-024-01991-z","DOIUrl":"10.1007/s00205-024-01991-z","url":null,"abstract":"<div><p>We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (in: International mathematical congress, Heidelberg, 1904; see Gesammelte Abhandlungen II, 1961) and Batchelor (J Fluid Mech 1:177–190, 1956), any Euler solution arising in this limit and consisting of a single “eddy” must have constant vorticity. Feynman and Lagerstrom (in: Proceedings of IX international congress on applied mechanics, 1956) gave a procedure to select the value of this vorticity by demanding a <i>necessary</i> condition for the existence of a periodic Prandtl boundary layer description. In the case of the disc, the choice—known to Batchelor (1956) and Wood (J Fluid Mech 2:77–87, 1957)—is explicit in terms of the slip forcing. For domains with non-constant curvature, Feynman and Lagerstrom give an approximate formula for the choice which is in fact only implicitly defined and must be determined together with the boundary layer profile. We show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. Due to the quasilinear coupling between the solution and the selected vorticity, we devise a delicate iteration scheme coupled with a high-order energy method that captures and controls the implicit selection mechanism.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195489","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

A Maximisation Technique for Solitary Waves: The Case of the Nonlocally Dispersive Whitham Equation 孤波的最大化技术：非局部分散惠瑟姆方程案例

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-05-29 DOI: 10.1007/s00205-024-01998-6

Mathias Nikolai Arnesen, Mats Ehrnström, Atanas G. Stefanov

{"title":"A Maximisation Technique for Solitary Waves: The Case of the Nonlocally Dispersive Whitham Equation","authors":"Mathias Nikolai Arnesen, Mats Ehrnström, Atanas G. Stefanov","doi":"10.1007/s00205-024-01998-6","DOIUrl":"10.1007/s00205-024-01998-6","url":null,"abstract":"<div><p>Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a different approach by maximising directly the dispersive part of the energy functional, while keeping the remaining nonlinear terms fixed with an Orlicz-space constraint. This method is, to the best of our knowledge new in the setting of water waves. The constructed solutions are bell-shaped in the sense that they are even, one-sided monotone, and attain their maximum at the origin. The method initially considers weaker solutions than in earlier works, and is not limited to small waves: a family of solutions is obtained, along which the dispersive energy is continuous and increasing. In general, our construction admits more than one solution for each energy level, and waves with the same energy level may have different heights. Although a transformation in the construction hinders us from concluding the family with an extreme wave, we give a quantitative proof that the set reaches ‘large’ or ‘intermediate-sized’ waves.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-01998-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Invariant Manifolds of Degenerate Tori and Double Parabolic Orbits to Infinity in the ((n+2))-Body Problem 在$$(n+2)$$-体问题中，畸变环和双抛物线轨道到无穷远的不变曲率

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-05-28 DOI: 10.1007/s00205-024-01995-9

Inmaculada Baldomá, Ernest Fontich, Pau Martín

引用次数: 0