Archive for Rational Mechanics and Analysis最新文献

{"title":"Well-Posedness of Degenerate Initial-Boundary Value Problems to a Hyperbolic-Parabolic Coupled System Arising from Nematic Liquid Crystals","authors":"Yanbo Hu,&nbsp;Yuusuke Sugiyama","doi":"10.1007/s00205-025-02093-0","DOIUrl":"10.1007/s00205-025-02093-0","url":null,"abstract":"<div><p>This paper is focused on the local well-posedness of initial-boundary value and Cauchy problems to a one-dimensional quasilinear hyperbolic-parabolic coupled system with boundary or far field degenerate initial data. The governing system is derived from the theory of nematic liquid crystals, which couples a hyperbolic equation describing the crystal property and a parabolic equation describing the liquid property of the material. The hyperbolic equation is degenerate at the boundaries or spatial infinity, which results in the classical methods for the strictly hyperbolic-parabolic coupled systems being invalid. We introduce admissible weighted function spaces and apply the parametrix method to construct iteration mappings for these two degenerate problems separately. The local existence and uniqueness of classical solutions of the degenerate initial-boundary value and Cauchy problems are established by the contraction mapping principle in their selected function spaces. Moreover, the solutions have no loss of regularity and their existence times are independent of the spatial variable.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143513360","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}

{"title":"Quantitative Homogenization of State-Constraint Hamilton–Jacobi Equations on Perforated Domains and Applications","authors":"Yuxi Han,&nbsp;Wenjia Jing,&nbsp;Hiroyoshi Mitake,&nbsp;Hung V. Tran","doi":"10.1007/s00205-025-02091-2","DOIUrl":"10.1007/s00205-025-02091-2","url":null,"abstract":"<div><p>We study the periodic homogenization problem of state-constraint Hamilton–Jacobi equations on perforated domains in the convex setting and obtain the optimal convergence rate. We then consider a dilute situation in which the diameter of the holes is much smaller than the microscopic scale. Finally, a homogenization problem with domain defects where some holes are missing is analyzed.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143489387","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}

{"title":"Local Decay Estimates","authors":"Avy Soffer,&nbsp;Xiaoxu Wu","doi":"10.1007/s00205-025-02089-w","DOIUrl":"10.1007/s00205-025-02089-w","url":null,"abstract":"<div><p>We give a proof of local decay estimates for Schrödinger-type equations, which is based on the knowledge of Asymptotic Completeness. This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Estimates or related methods. Global in time Strichartz estimates follow for quasi-periodic time-dependent potentials from our results.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02089-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396694","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}

{"title":"On the Minimization of the Willmore Energy Under a Constraint on Total Mean Curvature and Area","authors":"Christian Scharrer,&nbsp;Alexander West","doi":"10.1007/s00205-025-02087-y","DOIUrl":"10.1007/s00205-025-02087-y","url":null,"abstract":"<div><p>Motivated by a model for lipid bilayer cell membranes, we study the minimization of the Willmore functional in the class of oriented closed surfaces with prescribed total mean curvature, prescribed area, and prescribed genus. Adapting methods previously developed by Keller–Mondino–Rivière, Bauer–Kuwert, and Ndiaye–Schätzle, we prove the existence of smooth minimizers for a large class of constraints. Moreover, we analyze the asymptotic behaviour of the energy profile close to the unit sphere and consider the total mean curvature of axisymmetric surfaces.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02087-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143404265","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}

{"title":"Minimizers for an Aggregation Model with Attractive–Repulsive Interaction","authors":"Rupert L. Frank,&nbsp;Ryan W. Matzke","doi":"10.1007/s00205-025-02084-1","DOIUrl":"10.1007/s00205-025-02084-1","url":null,"abstract":"<div><p>We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that in an optimal part of the remaining parameter regime all minimizers are uniform distributions on a surface of a sphere, thus showing concentration on a lower dimensional set. Our method of proof uses convexity estimates on hypergeometric functions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02084-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388743","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}

{"title":"Minimality of Vortex Solutions to Ginzburg–Landau Type Systems for Gradient Fields in the Unit Ball in Dimension (Nge 4)","authors":"Radu Ignat,&nbsp;Mickael Nahon,&nbsp;Luc Nguyen","doi":"10.1007/s00205-025-02082-3","DOIUrl":"10.1007/s00205-025-02082-3","url":null,"abstract":"<div><p>We prove that the degree-one vortex solution is the unique minimizer for the Ginzburg–Landau functional for gradient fields (that is, the Aviles–Giga model) in the unit ball <span>(B^N)</span> in dimension <span>(N ge 4)</span> and with respect to its boundary value. A similar result is also prove in a model for <span>(mathbb {S}^N)</span>-valued maps arising in the theory of micromagnetics. Two methods are presented. The first method is an extension of the analogous technique previously used to treat the unconstrained Ginzburg–Landau functional in dimension <span>(N ge 7)</span>. The second method uses a symmetrization procedure for gradient fields such that the <span>(L^2)</span>-norm is invariant while the <span>(L^p)</span>-norm with <span>(2&lt; p &lt; infty )</span> and the <span>(H^1)</span>-norm are lowered.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02082-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109642","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}

{"title":"Traveling Front Solutions of Dimension n Generate Entire Solutions of Dimension ((n-1)) in Reaction–Diffusion Equations as the Speeds Go to Infinity","authors":"Hirokazu Ninomiya,&nbsp;Masaharu Taniguchi","doi":"10.1007/s00205-025-02083-2","DOIUrl":"10.1007/s00205-025-02083-2","url":null,"abstract":"<div><p>Multidimensional traveling front solutions and entire solutions of reaction–diffusion equations have been studied intensively. To study the relationship between multidimensional traveling front solutions and entire solutions, we study the reaction–diffusion equation with a bistable nonlinear term. It is well known that there exist multidimensional traveling front solutions with every speed that is greater than the speed of a one-dimensional traveling front solution connecting two stable equilibria. In this paper, we show that the limit of the <i>n</i>-dimensional multidimensional traveling front solutions as the speeds go to infinity generates an entire solution of the same reaction–diffusion equation in the <span>((n-1))</span>-dimensional space.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02083-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995539","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}

{"title":"Grand-Canonical Optimal Transport","authors":"Simone Di Marino,&nbsp;Mathieu Lewin,&nbsp;Luca Nenna","doi":"10.1007/s00205-024-02080-x","DOIUrl":"10.1007/s00205-024-02080-x","url":null,"abstract":"<div><p>We study a generalization of the multi-marginal optimal transport problem, which has no fixed number of marginals <i>N</i> and is inspired of statistical mechanics. It consists in optimizing a linear combination of the costs for all the possible <i>N</i>’s, while fixing a certain linear combination of the corresponding marginals.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142976459","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}

{"title":"Regularity and Nondegeneracy for Tumor Growth with Nutrients","authors":"Carson Collins,&nbsp;Matt Jacobs,&nbsp;Inwon Kim","doi":"10.1007/s00205-024-02081-w","DOIUrl":"10.1007/s00205-024-02081-w","url":null,"abstract":"<div><p>In this paper, we study a tumor growth model where the growth is driven by a diffusing nutrient and the tumor expands according to Darcy’s law with a mechanical pressure resulting from the incompressibility of the cells. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes. A crucial element in our analysis is establishing the regularity of the <i>hitting time</i> <i>T</i>(<i>x</i>), namely the first time the tumor patch reaches a given point. We achieve this by introducing a novel Hamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is of independent interest. The HJB structure is obtained by viewing the model as a limit of the Porous Media Equation (PME) and building upon a new variant of the AB estimate. Using the HJB structure, we establish a new Hopf-Lax type formula for the pressure variable. Combined with barrier arguments, the formula allows us to show that <i>T</i> is <span>(C^{alpha })</span> with <span>(alpha =alpha (d))</span>, which translates into a mild nondegeneracy of the tumor patch evolution. Building on this and obstacle problem theory, we show that the tumor patch boundary is regular in <span>({ mathbb {R}}^dtimes (0,infty ))</span> except on a set of Hausdorff dimension at most <span>(d-alpha )</span>. On the set of regular points, we further show that the tumor patch is locally <span>(C^{1,alpha })</span> in space-time. This conclusively establishes that instabilities in the boundary evolution do not amplify arbitrarily high frequencies.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142941128","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}

{"title":"Remarkable Localized Integral Identities for 3D Compressible Euler Flow and the Double-Null Framework","authors":"Leonardo Abbrescia,&nbsp;Jared Speck","doi":"10.1007/s00205-024-01997-7","DOIUrl":"10.1007/s00205-024-01997-7","url":null,"abstract":"<div><p>We derive new, localized geometric integral identities for solutions to the 3<i>D</i> compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The integral identities are coercive in the derivatives of the specific vorticity (defined to be vorticity divided by density) and the derivatives of the entropy gradient vectorfield, and the error terms exhibit remarkable regularity and null structures. Our framework plays a fundamental role in our companion works (Abbrescia L, Speck J. The emergence of the singular boundary from the crease in 3<i>D</i> compressible Euler flow, 2022; Abbrescia and Speck, The emergence of the Cauchy horizon from the crease in 3<i>D</i> compressible Euler flow (in preparation)) on the structure of the maximal classical development for shock-forming solutions. It allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions, and our approach reveals fundamental new coordinate-invariant structural features of the flow. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution’s higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: (i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric, where the top and bottom boundaries are acoustically spacelike—but not necessarily equal to portions of constant Cartesian-time hypersurfaces; and (ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. The main new contribution of the present article is our analysis of the spacelike, co-dimension one and two boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation and using careful geometric decompositions, we observe several crucial cancellations, which in total show that after a further integration with respect to an acoustical time function, the boundary integrals have a good sign, up to error terms that can be controlled due to their good null structure and regularity properties.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-01997-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142880508","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}