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Covariance-Modulated Optimal Transport and Gradient Flows 协方差调制优化传输和梯度流动
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-12-03 DOI: 10.1007/s00205-024-02065-w
Martin Burger, Matthias Erbar, Franca Hoffmann, Daniel Matthes, André Schlichting
{"title":"Covariance-Modulated Optimal Transport and Gradient Flows","authors":"Martin Burger,&nbsp;Matthias Erbar,&nbsp;Franca Hoffmann,&nbsp;Daniel Matthes,&nbsp;André Schlichting","doi":"10.1007/s00205-024-02065-w","DOIUrl":"10.1007/s00205-024-02065-w","url":null,"abstract":"<div><p>We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02065-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844752","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 Structures for Quasilinear Singular SPDEs 准线性奇异 SPDE 的正则结构
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-29 DOI: 10.1007/s00205-024-02069-6
I. Bailleul, M. Hoshino, S. Kusuoka
{"title":"Regularity Structures for Quasilinear Singular SPDEs","authors":"I. Bailleul,&nbsp;M. Hoshino,&nbsp;S. Kusuoka","doi":"10.1007/s00205-024-02069-6","DOIUrl":"10.1007/s00205-024-02069-6","url":null,"abstract":"<div><p>We prove the well-posed character of a regularity structure formulation of the quasilinear generalized (KPZ) equation and give an explicit form for a renormalized equation in the full subcritical regime. Under the assumption that the BPHZ models associated with a non-translation invariant operator converge, we obtain a convergence result for the solutions of the regularized renormalized equations. This conditional result covers the spacetime white noise case.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142737043","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
Potential Theory for Nonlocal Drift-Diffusion Equations 非局部漂移-扩散方程的势理论
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-29 DOI: 10.1007/s00205-024-02073-w
Quoc-Hung Nguyen, Simon Nowak, Yannick Sire, Marvin Weidner
{"title":"Potential Theory for Nonlocal Drift-Diffusion Equations","authors":"Quoc-Hung Nguyen,&nbsp;Simon Nowak,&nbsp;Yannick Sire,&nbsp;Marvin Weidner","doi":"10.1007/s00205-024-02073-w","DOIUrl":"10.1007/s00205-024-02073-w","url":null,"abstract":"<div><p>The purpose of this paper is to prove new fine regularity results for nonlocal drift-diffusion equations via pointwise potential estimates. Our analysis requires only minimal assumptions on the divergence free drift term, enabling us to include drifts of critical order belonging merely to BMO. In particular, our results allow us to derive new estimates for the dissipative surface quasi-geostrophic equation.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142737189","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
Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws 超曲线正则化守恒律系统冲击剖面的谱稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-26 DOI: 10.1007/s00205-024-02066-9
Johannes Bärlin
{"title":"Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws","authors":"Johannes Bärlin","doi":"10.1007/s00205-024-02066-9","DOIUrl":"10.1007/s00205-024-02066-9","url":null,"abstract":"<div><p>We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable if the shock amplitude is sufficiently small. This means that an associated Evans function <span>(mathcal {E}:Lambda rightarrow mathbb {C})</span> with <span>(Lambda subset mathbb {C})</span> an open superset of the closed right half plane <span>(mathbb {H}^+equiv {kappa in mathbb {C}:text {Re},kappa geqq 0})</span> has only one zero, namely, a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan (Arch Ration Mech Anal 164:287–309, 2002) and Plaza and Zumbrun (Discrete Contin Dyn Syst 10(4):885–924, 2004) for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun (Partial Differ Equ 34(1–3):119–136, 2009), Plaza and Zumbrun (2004) and Ueda (Math Methods Appl Sci 32(4):419–434, 2009).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02066-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714502","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
A New Condition on the Vorticity for Partial Regularity of a Local Suitable Weak Solution to the Navier–Stokes Equations 纳维-斯托克斯方程局部合适弱解的涡度部分正则性新条件
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-25 DOI: 10.1007/s00205-024-02068-7
Dongho Chae, Jörg Wolf
{"title":"A New Condition on the Vorticity for Partial Regularity of a Local Suitable Weak Solution to the Navier–Stokes Equations","authors":"Dongho Chae,&nbsp;Jörg Wolf","doi":"10.1007/s00205-024-02068-7","DOIUrl":"10.1007/s00205-024-02068-7","url":null,"abstract":"<div><p>We provide a new <span>(varepsilon )</span>-condition for the vorticity of a suitable weak solution to the Navier–Stokes equations that leads to partial regularity. This refines the well known limsup condition of the Caffarelli-Kohn-Nirenberg Theorem by a new condition on the vorticity, replacing limsup by a suitable range of the radius <i>r</i> of the parabolic cylinders. As a consequence, the partial regularity is obtained directly from this <span>(varepsilon )</span>-condition of the vorticity without relying on the <span>(varepsilon )</span>-condition of the velocity. Furthermore, by the local nature of the method this result holds for any local suitable weak solution of the Navier–Stokes equations in a general domain.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714239","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
Boundary Behavior of Limit-Interfaces for the Allen–Cahn Equation on Riemannian Manifolds with Neumann Boundary Condition 具有诺伊曼边界条件的黎曼曼体上艾伦-卡恩方程的极限界面的边界行为
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-25 DOI: 10.1007/s00205-024-02070-z
Martin Man-chun Li, Davide Parise, Lorenzo Sarnataro
{"title":"Boundary Behavior of Limit-Interfaces for the Allen–Cahn Equation on Riemannian Manifolds with Neumann Boundary Condition","authors":"Martin Man-chun Li,&nbsp;Davide Parise,&nbsp;Lorenzo Sarnataro","doi":"10.1007/s00205-024-02070-z","DOIUrl":"10.1007/s00205-024-02070-z","url":null,"abstract":"<div><p>We study the boundary behavior of any limit-interface arising from a sequence of general critical points of the Allen–Cahn energy functionals on a smooth bounded domain. Given any such sequence with uniform energy bounds, we prove that the limit-interface is a free boundary varifold which is integer rectifiable up to the boundary. This extends earlier work of Hutchinson and Tonegawa on the interior regularity of the limit-interface. A key novelty in our result is that no convexity assumption of the boundary is required and it is valid even when the limit-interface clusters near the boundary. Moreover, our arguments are local and thus work in the Riemannian setting. This work provides the first step towards the regularity theory for the Allen–Cahn min-max theory for free boundary minimal hypersurfaces, which was developed in the Almgren–Pitts setting by the first-named author and Zhou.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02070-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714240","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
Low Regularity Solutions for the General Quasilinear Ultrahyperbolic Schrödinger Equation 一般准线性超双曲薛定谔方程的低正则性解决方案
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-23 DOI: 10.1007/s00205-024-02072-x
Ben Pineau, Mitchell A. Taylor
{"title":"Low Regularity Solutions for the General Quasilinear Ultrahyperbolic Schrödinger Equation","authors":"Ben Pineau,&nbsp;Mitchell A. Taylor","doi":"10.1007/s00205-024-02072-x","DOIUrl":"10.1007/s00205-024-02072-x","url":null,"abstract":"<div><p>We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schrödinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive improvement over the landmark results of Kenig, Ponce, Rolvung and Vega (Kenig et al. in Adv Math 196:373–486, 2005; Carlos et al. in Adv Math 206:402–433, 2006; Carlos et al. in Invent Math 134:489–545, 1998; Carlos et al. in Invent Math 158:343–388, 2004), as it weakens the regularity and decay assumptions to the same scale of spaces considered by Marzuola, Metcalfe and Tataru in (Jeremy et al. in Arch Ration Mech Anal 242:1119-1175, 2021), but removes the uniform ellipticity assumption on the metric from their result. Our method has the additional benefit of being relatively simple but also very robust. In particular, it only relies on the use of pseudodifferential calculus for classical symbols.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142691872","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
Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces 沉入稳定超曲面的正切圆锥的唯一性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-22 DOI: 10.1007/s00205-024-02071-y
Nick Edelen, Paul Minter
{"title":"Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces","authors":"Nick Edelen,&nbsp;Paul Minter","doi":"10.1007/s00205-024-02071-y","DOIUrl":"10.1007/s00205-024-02071-y","url":null,"abstract":"<div><p>We establish uniqueness and regularity results for tangent cones (at a point or at infinity), with isolated singularities arising from a given immersed stable minimal hypersurface with suitably small (non-immersed) singular set. In particular, our results allow the tangent cone to occur with any integer multiplicity.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02071-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679644","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
Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars 被动标量中的规范增长、非唯一性和异常耗散
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-21 DOI: 10.1007/s00205-024-02056-x
Tarek M. Elgindi, Kyle Liss
{"title":"Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars","authors":"Tarek M. Elgindi,&nbsp;Kyle Liss","doi":"10.1007/s00205-024-02056-x","DOIUrl":"10.1007/s00205-024-02056-x","url":null,"abstract":"<div><p>We construct a divergence-free velocity field <span>(u:[0,T] times mathbb {T}^2 rightarrow mathbb {R}^2)</span> satisfying </p><div><div><span>$$u in C^infty ([0,T];C^alpha (mathbb {T}^2)) quad forall alpha in [0,1)$$</span></div></div><p>such that the corresponding drift-diffusion equation exhibits anomalous dissipation for all smooth initial data. We also show that, given any <span>(alpha _0 &lt; 1)</span>, the flow can be modified such that it is uniformly bounded only in <span>(C^{alpha _0}(mathbb {T}^2))</span> and the regularity of solutions satisfy sharp (time-integrated) bounds predicted by the Obukhov–Corrsin theory. The proof is based on a general principle implying <span>(H^1)</span> growth for all solutions to the transport equation, which may be of independent interest.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679772","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 the Characterization, Existence and Uniqueness of Steady Solutions to the Hydrostatic Euler Equations in a Nozzle 论喷嘴中静水欧拉方程稳定解的特征、存在性和唯一性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-20 DOI: 10.1007/s00205-024-02062-z
Wang Shing Leung, Tak Kwong Wong, Chunjing Xie
{"title":"On the Characterization, Existence and Uniqueness of Steady Solutions to the Hydrostatic Euler Equations in a Nozzle","authors":"Wang Shing Leung,&nbsp;Tak Kwong Wong,&nbsp;Chunjing Xie","doi":"10.1007/s00205-024-02062-z","DOIUrl":"10.1007/s00205-024-02062-z","url":null,"abstract":"<div><p>Incompressible Euler flows in narrow domains, in which the horizontal length scale is much larger than other scales, play an important role in many different applications, and their leading-order behavior can be described by the hydrostatic Euler equations. In this paper, we show that steady solutions of the hydrostatic Euler equations in an infinite strip strictly away from stagnation must be shear flows. Furthermore, we prove the existence, uniqueness, and asymptotic behavior of global steady solutions to the hydrostatic Euler equations in general nozzles. In terms of stream function formulation, the hydrostatic Euler equations can be written as a degenerate elliptic equation, for which the Liouville type theorem in a strip is a consequence of the analysis for the second order ordinary differential equation (ODE). The analysis on the associated ODE also helps determine the far field behavior of solutions in general nozzles, which plays an important role in guaranteeing the equivalence of stream function formulation. One of the key ingredients for the analysis on flows in a general nozzle is a new transformation, which combines a change of variable and an Euler–Lagrange transformation. With the aid of this new transformation, the solutions in the new coordinates enjoy explicit representations so that the regularity with respect to the horizontal variable can be gained in a clear way.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142672491","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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