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Kac’s Program for the Landau Equation 兰道方程的卡茨程序
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-30 DOI: 10.1007/s00205-026-02230-3
Xuanrui Feng, Zhenfu Wang
{"title":"Kac’s Program for the Landau Equation","authors":"Xuanrui Feng,&nbsp;Zhenfu Wang","doi":"10.1007/s00205-026-02230-3","DOIUrl":"10.1007/s00205-026-02230-3","url":null,"abstract":"<div><p>We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative <i>N</i>-particle system, obtained by passing to the grazing limit on Kac’s walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of the propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac’s program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses, together with strong <span>(L^1)</span> convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch–Duerinckx–Jabin [5] and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614939","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 (L^p) Regularity Problem for Parabolic Operators 抛物算子的(L^p)正则性问题
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-25 DOI: 10.1007/s00205-026-02212-5
Martin Dindoš, Linhan Li, Jill Pipher
{"title":"The (L^p) Regularity Problem for Parabolic Operators","authors":"Martin Dindoš,&nbsp;Linhan Li,&nbsp;Jill Pipher","doi":"10.1007/s00205-026-02212-5","DOIUrl":"10.1007/s00205-026-02212-5","url":null,"abstract":"<div><p>In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE <span>(-partial _tu + textrm{div}(Anabla u)=0)</span> on a Lipschitz cylinder <span>(mathcal {O}times mathbb {R})</span> is solvable for some <span>(pin (1,infty ))</span> under the assumption that the matrix <i>A</i> is elliptic, has bounded and measurable coefficients, and that its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition).</p><p>We prove that for some <span>(p_0&gt;1)</span> the Regularity problem is solvable in the range <span>((1,p_0))</span>. We note that answer to this question was not known even in the <i>small Carleson case</i>, that is, when the Carleson norm of coefficients is sufficiently small.</p><p>In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614172","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
Correction to: Rigidity and Functional Properties of (text {BD}_{text {dev}}(Omega ))BDdev((Omega )) 修正：(text {BD}_{text {dev}}(Omega )) BDdev（(Omega )）的刚度和功能特性
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-22 DOI: 10.1007/s00205-026-02225-0
Marco Caroccia, Nicolas Van Goethem
{"title":"Correction to: Rigidity and Functional Properties of (text {BD}_{text {dev}}(Omega ))BDdev((Omega ))","authors":"Marco Caroccia,&nbsp;Nicolas Van Goethem","doi":"10.1007/s00205-026-02225-0","DOIUrl":"10.1007/s00205-026-02225-0","url":null,"abstract":"","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02225-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148613741","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
Linear and Nonlinear Phase Mixing for the Gravitational Vlasov–Poisson System Under an External Kepler Potential 外部开普勒势下引力Vlasov-Poisson系统的线性和非线性相位混合
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-18 DOI: 10.1007/s00205-026-02209-0
Sanchit Chaturvedi, Jonathan Luk
{"title":"Linear and Nonlinear Phase Mixing for the Gravitational Vlasov–Poisson System Under an External Kepler Potential","authors":"Sanchit Chaturvedi,&nbsp;Jonathan Luk","doi":"10.1007/s00205-026-02209-0","DOIUrl":"10.1007/s00205-026-02209-0","url":null,"abstract":"<div><p>In Newtonian gravity, a self-gravitating collisionless gas around a massive object such as a star or a planet is modeled via the Vlasov–Poisson system with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object nor escape to infinity. We study this system focusing on the regime with bounded trajectories. First, we prove quantitative linear phase mixing estimates in three dimensions outside symmetry. Second, our main result is a long-time nonlinear phase mixing theorem for spherically symmetric data with finite regularity. The mechanism is phenomenologically similar to Landau damping on a torus and our result applies to the same time scale (modulo logarithms) as the known results on Landau damping with finite regularity. However, in contrast with Landau damping, we need to contend with weaker linear estimates as well as use a system of dynamically defined action angle variables.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148466365","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
Quantum Systems at The Brink: Helium–Type Systems 边缘的量子系统：氦型系统。
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-13 DOI: 10.1007/s00205-026-02167-7
Dirk Hundertmark, Michal Jex, Markus Lange
{"title":"Quantum Systems at The Brink: Helium–Type Systems","authors":"Dirk Hundertmark,&nbsp;Michal Jex,&nbsp;Markus Lange","doi":"10.1007/s00205-026-02167-7","DOIUrl":"10.1007/s00205-026-02167-7","url":null,"abstract":"<div><p>In the present paper we study two challenging problems for helium–type systems: the existence of eigenvalues at thresholds and the asymptotic behavior of the corresponding eigenfunctions. Since the usual methods for addressing these problems need a safety distance to the essential spectrum, they cannot be applied in critical cases, when an eigenvalue enters the continuum. We develop a method to address both problems and derive <i>sharp upper and lower bounds</i> for the asymptotic behavior of the ground state of critical helium–type systems at the threshold of the essential spectrum. This is the first proof of the precise asymptotic behavior of the ground state for this benchmark problem in quantum chemistry. Moreover, our bounds describe precisely how the asymptotic decay of the ground state changes, when the system becomes critical.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13364814/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148450715","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
Stability of Rayleigh–Jeans Equilibria in the Kinetic FPU Equation 动力学FPU方程中Rayleigh-Jeans平衡的稳定性。
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-07 DOI: 10.1007/s00205-026-02223-2
Pierre Germain, Joonhyun La, Angeliki Menegaki
{"title":"Stability of Rayleigh–Jeans Equilibria in the Kinetic FPU Equation","authors":"Pierre Germain,&nbsp;Joonhyun La,&nbsp;Angeliki Menegaki","doi":"10.1007/s00205-026-02223-2","DOIUrl":"10.1007/s00205-026-02223-2","url":null,"abstract":"<div><p>We study the nonlinear dynamics of the kinetic wave equation associated with the FPU problem and prove the stability of the non-singular Rayleigh–Jeans equilibria. The lack of a spectral gap for the linearized problem leads to polynomial decay, which we are able to leverage to obtain nonlinear stability.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13342143/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148413896","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
Local Energy Control in the Presence of a Zero-Energy Resonance 零能量共振存在下的局部能量控制
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-07 DOI: 10.1007/s00205-026-02220-5
José M. Palacios, Fabio Pusateri
{"title":"Local Energy Control in the Presence of a Zero-Energy Resonance","authors":"José M. Palacios,&nbsp;Fabio Pusateri","doi":"10.1007/s00205-026-02220-5","DOIUrl":"10.1007/s00205-026-02220-5","url":null,"abstract":"<div><p>We consider the problem of stability and local energy decay for co-dimension one perturbations of the soliton of the cubic Klein–Gordon equation in <span>(1+1)</span> dimensions. Our main result gives a weighted time-averaged control of the local energy over a time interval which is exponentially long in the size of the initial (total) energy. More precisely, for well-prepared initial perturbations on the center stable manifold that are of size <span>(delta )</span> in the energy norm, we show that the local energy is under control up to times of the order <span>(exp (cdelta ^{-beta }))</span> for any <span>(beta &lt; 4/3)</span>. A major difficulty is the presence of a zero-energy resonance in the linearized operator, which is a well-known obstruction to improved local decay properties. We address this issue by using virial estimates that are frequency localized in a time-dependent way, and introducing a “singular virial functional” with time dependent weights to control the mass of the perturbation projected away from small frequencies. The proof applies to more general models, yielding analogous results for perturbations of the kink of the Sine-Gordon model, and small solution of nonlinear Klein–Gordon equations. In this respect, our result is close to optimal due to the existence of wobbling kinks and breathers in the Sine-Gordon model which violate our conclusion if <span>(beta = 2)</span>. This appears to be the first successful general attempt at using virial estimates in the presence of a resonance to deduce local energy control.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148465105","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
Asymptotic Stability of the Three-Dimensional Couette Flow for the Stokes-Transport Equations stokes -输运方程三维Couette流的渐近稳定性
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-02 DOI: 10.1007/s00205-026-02219-y
Daniel Sinambela, Weiren Zhao, Ruizhao Zi
{"title":"Asymptotic Stability of the Three-Dimensional Couette Flow for the Stokes-Transport Equations","authors":"Daniel Sinambela,&nbsp;Weiren Zhao,&nbsp;Ruizhao Zi","doi":"10.1007/s00205-026-02219-y","DOIUrl":"10.1007/s00205-026-02219-y","url":null,"abstract":"<div><p>In this paper, we investigate the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equations. We observe that a lift-up effect similar to that in the three-dimensional Navier–Stokes equation near Couette flow destabilizes the system, while the inviscid-damping-type decay due to the Couette flow <span>((Y,0,0)^top )</span>, together with the damping structure induced by the decreasing background density <span>(varrho _textrm{s}(Y))</span>, stabilizes the system. More precisely, we prove that if the initial density is close to a linearly decreasing function in the Gevrey-<span>(frac{1}{s})</span> class with <span>(frac{1}{2}&lt; sleqq 1)</span>, namely, <span>(Vert varrho _textrm{in}(X,Y,Z)-varrho _textrm{s}(Y)Vert _{mathcal {G}^{s}}leqq varepsilon )</span>, then the perturbed density remains close to <span>(varrho _textrm{s}(Y))</span>. Moreover, the associated velocity field converges to Couette flow <span>((Y, 0, 0)^{top })</span> with a convergence rate of <span>(frac{1}{langle trangle ^3})</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148380237","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
Correction: Maximal Dissipation and Well-Posedness of the Euler System of Gas Dynamics 修正：气体动力学欧拉系统的最大耗散和适定性
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-02 DOI: 10.1007/s00205-026-02222-3
Eduard Feireisl, Ansgar Jüngel, Mária Lukáčová-Medviďová
{"title":"Correction: Maximal Dissipation and Well-Posedness of the Euler System of Gas Dynamics","authors":"Eduard Feireisl,&nbsp;Ansgar Jüngel,&nbsp;Mária Lukáčová-Medviďová","doi":"10.1007/s00205-026-02222-3","DOIUrl":"10.1007/s00205-026-02222-3","url":null,"abstract":"","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02222-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148380238","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
Existence and Partial Regularity of Legendrian Area-Minimizing Currents Legendrian最小面积电流的存在性和部分正则性
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-02 DOI: 10.1007/s00205-026-02186-4
Gerard Orriols
{"title":"Existence and Partial Regularity of Legendrian Area-Minimizing Currents","authors":"Gerard Orriols","doi":"10.1007/s00205-026-02186-4","DOIUrl":"10.1007/s00205-026-02186-4","url":null,"abstract":"<div><p>We show that Legendrian integral currents in a contact manifold that locally minimize the mass among Legendrian competitors have a regular set which is open and dense in their support. We apply this to show the existence and partial regularity of solutions of the Legendrian Plateau problem in the <i>n</i>th Heisenberg group for an arbitrary horizontal <span>((n-1))</span>-cycle as prescribed boundary, and of mass-minimizing Legendrian integral currents in any <i>n</i>-dimensional homology class of a closed contact <span>((2n+1))</span>-manifold. In the case of the Heisenberg group, our result applies to Ambrosio–Kirchheim metric currents with respect to the Carnot–Carathéodory distance. Our results do not assume any compatibility between the subriemannian metric and the symplectic form.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02186-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148380670","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
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