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Damping Versus Oscillations for a Gravitational Vlasov–Poisson System 引力Vlasov-Poisson系统的阻尼与振荡。
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-07-17 DOI: 10.1007/s00205-025-02114-y
M. Hadžić, G. Rein, M. Schrecker, C. Straub
{"title":"Damping Versus Oscillations for a Gravitational Vlasov–Poisson System","authors":"M. Hadžić,&nbsp;G. Rein,&nbsp;M. Schrecker,&nbsp;C. Straub","doi":"10.1007/s00205-025-02114-y","DOIUrl":"10.1007/s00205-025-02114-y","url":null,"abstract":"<div><p>We consider a family of isolated inhomogeneous steady states of the gravitational Vlasov–Poisson system with a point mass at the centre. These are parametrised by the polytropic index <span>(k&gt;1/2)</span>, so that the phase space density of the steady state is <span>(C^1)</span> at the vacuum boundary if and only if <span>(k&gt;1)</span>. We prove the following sharp dichotomy result: if <span>(k&gt;1)</span>, the linear perturbations Landau damp and if <span>(1/2&lt; kle 1)</span> they do not. The above dichotomy is a new phenomenon and highlights the importance of steady state regularity at the vacuum boundary in the discussion of the long-time behaviour of the perturbations. Our proof of (nonquantitative) gravitational relaxation around steady states with <span>(k&gt;1)</span> is the first such result for the gravitational Vlasov–Poisson system. The key novelty of this work is the proof that no embedded eigenvalues exist in the essential spectrum of the linearised system.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12271275/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144676636","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
The Excitation Spectrum of a Bose Gas with an Impurity in the Gross–Pitaevskii Regime 含杂质玻色气体在Gross-Pitaevskii体系中的激发谱
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-07-03 DOI: 10.1007/s00205-025-02112-0
Jonas Lampart, Arnaud Triay
{"title":"The Excitation Spectrum of a Bose Gas with an Impurity in the Gross–Pitaevskii Regime","authors":"Jonas Lampart,&nbsp;Arnaud Triay","doi":"10.1007/s00205-025-02112-0","DOIUrl":"10.1007/s00205-025-02112-0","url":null,"abstract":"<div><p>We study a dilute system of <i>N</i> interacting bosons coupled to an impurity particle via a pair potential in the Gross–Pitaevskii regime. We derive an expansion of the ground state energy up to order one in the boson number, and show that the difference of excited eigenvalues to the ground state is given by the eigenvalues of the renormalized Bogoliubov–Fröhlich Hamiltonian in the limit <span>(Nrightarrow infty )</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145161701","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
On a Class of Generalised Solutions to the Kinetic Hookean Dumbbell Model for Incompressible Dilute Polymeric Fluids: Existence and Macroscopic Closure 不可压缩稀聚合物流体动力学Hookean哑铃模型的一类广义解：存在性和宏观闭包性
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-07-02 DOI: 10.1007/s00205-025-02115-x
Tomasz Dębiec, Endre Süli
{"title":"On a Class of Generalised Solutions to the Kinetic Hookean Dumbbell Model for Incompressible Dilute Polymeric Fluids: Existence and Macroscopic Closure","authors":"Tomasz Dębiec,&nbsp;Endre Süli","doi":"10.1007/s00205-025-02115-x","DOIUrl":"10.1007/s00205-025-02115-x","url":null,"abstract":"<div><p>We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier–Stokes equations in a bounded Lipschitz domain, coupled to a Fokker–Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent. The micro–macro interaction is reflected by the presence of a drag term in the Fokker–Planck equation and the divergence of a polymeric extra-stress tensor in the Navier–Stokes balance of momentum equation. We introduce the concept of <i>generalised dissipative solution</i>—a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation. This defect measure accounts for the lack of compactness in the polymeric extra-stress tensor. We prove global existence of generalised dissipative solutions satisfying additionally an energy inequality for the macroscopic deformation tensor. Using this inequality, we establish a conditional regularity result: any generalised dissipative solution with a sufficiently regular velocity field is a weak solution to the Hookean dumbbell model. Additionally, in two space dimensions we provide a rigorous derivation of the macroscopic closure of the Hookean model and discuss its relationship with the Oldroyd-B model with stress diffusion. Finally, we improve a result by Barrett and Süli (Nonlinear Anal. Real World Appl. 39:362–395, 2018) by establishing the global existence of weak solutions for a larger class of initial data.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02115-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145160927","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
Time-Asymptotic Stability of Generic Riemann Solutions for Compressible Navier–Stokes–Fourier Equations 可压缩Navier-Stokes-Fourier方程一般Riemann解的时间渐近稳定性
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-06-25 DOI: 10.1007/s00205-025-02116-w
Moon-Jin Kang, Alexis F. Vasseur, Yi Wang
{"title":"Time-Asymptotic Stability of Generic Riemann Solutions for Compressible Navier–Stokes–Fourier Equations","authors":"Moon-Jin Kang,&nbsp;Alexis F. Vasseur,&nbsp;Yi Wang","doi":"10.1007/s00205-025-02116-w","DOIUrl":"10.1007/s00205-025-02116-w","url":null,"abstract":"<div><p>We establish the time-asymptotic stability of generic Riemann solutions to the one-dimensional compressible Navier–Stokes–Fourier equations. The Riemann solution under consideration is a generic combination of a shock, a contact discontinuity, and a rarefaction wave. We prove that the perturbed solution of Navier–Stokes–Fourier converges, uniformly in space as time goes to infinity, to an ansatz composed of viscous shock with time-dependent shift, a viscous contact wave and an inviscid rarefaction wave. This is a first resolution of the time-asymptotic stability of three waves of different kinds associated with the generic Riemann solutions. Our approach relies on the method of <i>a</i>-contraction with shifts and relative entropy, specifically applied to both the shock wave and the contact wave. It enables the application of a global energy method for the generic combination of three waves.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145168796","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
Long Time Stability of Hamiltonian Derivative Nonlinear Schrödinger Equations Without Potential 无势哈密顿导数非线性Schrödinger方程的长时间稳定性
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-06-06 DOI: 10.1007/s00205-025-02109-9
Hu Shengqing, Zhang Jing
{"title":"Long Time Stability of Hamiltonian Derivative Nonlinear Schrödinger Equations Without Potential","authors":"Hu Shengqing,&nbsp;Zhang Jing","doi":"10.1007/s00205-025-02109-9","DOIUrl":"10.1007/s00205-025-02109-9","url":null,"abstract":"<div><p>In this paper, we prove an abstract Birkhoff normal form theorem for some unbounded infinite dimensional Hamiltonian systems. Based on this result we obtain that the solution to Derivative Nonlinear Schrödinger equations under periodic boundary condition with typical small enough initial value remains small in the Sobolev norm <span>( H^{textbf{s}}(mathbb {T}))</span> over a long time interval. The length of the time interval is equal to <span>(e^{|ln R|^{1+gamma }})</span> with <span>(0&lt;gamma &lt;1/5)</span> as the initial value is smaller than <span>(Rll 1)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145162387","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
Long Time Validity of the Linearized Boltzmann Equation for Hard Spheres: A Proof Without Billiard Theory 硬球线性化玻尔兹曼方程的长时间有效性：不需要台球理论的证明
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-06-05 DOI: 10.1007/s00205-025-02105-z
Corentin Le Bihan
{"title":"Long Time Validity of the Linearized Boltzmann Equation for Hard Spheres: A Proof Without Billiard Theory","authors":"Corentin Le Bihan","doi":"10.1007/s00205-025-02105-z","DOIUrl":"10.1007/s00205-025-02105-z","url":null,"abstract":"<div><p>We study space–time fluctuations of a hard sphere system at thermal equilibrium, and prove that the covariance converges to the solution of a linearized Boltzmann equation in the low density limit, globally in time. This result was obtained previously in Bodineau et al. (Commun Pure Appl Math 76:3852–3911, 2021) by using uniform bounds on the number of recollisions of dispersing systems of hard spheres [as provided for instance in Burago et al. (Ann Math (2), 147(3):695–708, 1998)]. We present a self-contained proof with substantial differences, which does not use this geometric result. This can be regarded as the first step of a program aiming of deriving the fluctuation theory of the rarefied gas for interaction potentials different from hard spheres.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145162040","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
Construction of Fillings with Prescribed Gaussian Image and Applications 高斯图像填充的构造及其应用
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-06-02 DOI: 10.1007/s00205-025-02110-2
Antonio De Rosa, Yucong Lei, Robert Young
{"title":"Construction of Fillings with Prescribed Gaussian Image and Applications","authors":"Antonio De Rosa,&nbsp;Yucong Lei,&nbsp;Robert Young","doi":"10.1007/s00205-025-02110-2","DOIUrl":"10.1007/s00205-025-02110-2","url":null,"abstract":"<div><p>We construct <i>d</i>–dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the barycenter of the prescribed measure, considered as a measure on <span>(bigwedge ^d mathbb {R}^n)</span>, is 0), or their boundary is the boundary of a unit <i>d</i>–cube (if the barycenter of the prescribed measure is a simple <i>d</i>–vector). Such fillings were first proven to exist by Burago and Ivanov (Geom Funct Anal 14:469–490, 2004); our work gives an explicit construction, which is also flexible to generalizations. For instance, in the case that the measure on the Grassmannian is supported on the set of positively oriented <i>d</i>–planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove the surprising fact that, for anisotropic integrands, polyconvexity is equivalent to quasiconvexity of the associated <i>Q</i>-integrands (that is, ellipticity for Lipschitz multigraphs) and to show that strict polyconvexity is necessary for the atomic condition to hold.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145161036","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
Global Solutions of the One-Dimensional Compressible Euler Equations with Nonlocal Interactions via the Inviscid Limit 具有非局部相互作用的一维可压缩欧拉方程的无粘极限全局解
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-05-24 DOI: 10.1007/s00205-025-02097-w
José A. Carrillo, Gui-Qiang G. Chen, Difan Yuan, Ewelina Zatorska
{"title":"Global Solutions of the One-Dimensional Compressible Euler Equations with Nonlocal Interactions via the Inviscid Limit","authors":"José A. Carrillo,&nbsp;Gui-Qiang G. Chen,&nbsp;Difan Yuan,&nbsp;Ewelina Zatorska","doi":"10.1007/s00205-025-02097-w","DOIUrl":"10.1007/s00205-025-02097-w","url":null,"abstract":"<div><p>We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional compressible Euler equations with (possibly) damping, alignment forces, and the nonlocal interactions of Newtonian repulsion and quadratic confinement. Both the polytropic gas law and the general gas law are analyzed. This is achieved by constructing a sequence of solutions of the one-dimensional compressible Navier–Stokes-type equations with density-dependent viscosity on expanding intervals with the stress-free boundary condition and then taking the vanishing viscosity limit. The main difficulties in this paper arise from the appearance of the nonlocal terms. In particular, some uniform higher moment estimates of the solutions for the compressible Navier–Stokes equations on the expanding intervals with stress-free boundary condition are obtained by careful design of the approximate initial data.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02097-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144125867","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
The Discrete Dislocation Dynamics of Multiple Dislocation Loops 多位错环的离散位错动力学
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-05-20 DOI: 10.1007/s00205-025-02108-w
Stefania Patrizi, Mary Vaughan
{"title":"The Discrete Dislocation Dynamics of Multiple Dislocation Loops","authors":"Stefania Patrizi,&nbsp;Mary Vaughan","doi":"10.1007/s00205-025-02108-w","DOIUrl":"10.1007/s00205-025-02108-w","url":null,"abstract":"<div><p>We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls–Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in <span>(mathbb {R}^n)</span>, <span>(n ge 2)</span>. After suitably rescaling the equation with a small phase parameter <span>(varepsilon &gt;0)</span>, the rescaled solution solves a fractional Allen–Cahn equation. We show that, as <span>(varepsilon rightarrow 0)</span>, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144090982","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
Transverse Linear Stability of One-Dimensional Solitary Gravity Water Waves 一维孤立重力水波的横向线性稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-05-15 DOI: 10.1007/s00205-025-02101-3
Frédéric Rousset, Changzhen Sun
{"title":"Transverse Linear Stability of One-Dimensional Solitary Gravity Water Waves","authors":"Frédéric Rousset,&nbsp;Changzhen Sun","doi":"10.1007/s00205-025-02101-3","DOIUrl":"10.1007/s00205-025-02101-3","url":null,"abstract":"<div><p>In this paper, we establish the transverse linear asymptotic stability of one-dimensional small-amplitude solitary waves of the gravity water-waves system. More precisely, we show that the semigroup of the linearized operator about the solitary wave decays exponentially within a spectral subspace supplementary to the space generated by the spectral projection on continuous resonant modes. The key element of the proof is to establish suitable uniform resolvent estimates. To achieve this, we use different arguments depending on the size of the transverse frequencies. For high transverse frequencies, we use reductions based on pseudodifferential calculus, for intermediate ones, we use an energy-based approach relying on the design of various appropriate energy functionals for different regimes of longitudinal frequencies and for low frequencies, we use the KP-II approximation. As a corollary of our main result, we also get the spectral stability in the unweighted energy space.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143949458","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
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