Archive for Rational Mechanics and Analysis最新文献

{"title":"Stationary Self-Similar Profiles for the Two-Dimensional Inviscid Boussinesq Equations","authors":"Ken Abe,&nbsp;Daniel Ginsberg,&nbsp;In-Jee Jeong","doi":"10.1007/s00205-026-02195-3","DOIUrl":"10.1007/s00205-026-02195-3","url":null,"abstract":"<div><p>We consider (<span>(-alpha )</span>)-homogeneous solutions (stationary self-similar solutions of degree <span>(-alpha )</span>) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions. More specifically, we demonstrate the following:</p><ul>\u0000 <li>\u0000 <p>Non-existence of rotational (<span>(-alpha )</span>)-homogeneous solutions with regular profiles <span>((u,p,rho )in C^{1}(overline{mathbb {R}^{2}_{+}}backslash {0}))</span> for <span>(0leqq alpha leqq 1)</span> and <span>((u,p,rho )in C^{2}(overline{mathbb {R}^{2}_{+}}backslash {0}))</span> for <span>(-1/2leqq alpha &lt;0)</span>;</p>\u0000 </li>\u0000 <li>\u0000 <p>Existence of rotational (<span>(-alpha )</span>)-homogeneous solutions with regular profiles <span>((u,p,rho )in C^{2}(overline{mathbb {R}^{2}_{+}}backslash {0}))</span> for <span>(alpha &gt;1)</span> and <span>((u,p,rho )in C^{1}(overline{mathbb {R}^{2}_{+}}))</span> for <span>(alpha &lt;-2)</span>;</p>\u0000 </li>\u0000 <li>\u0000 <p>Existence of rotational (<span>(-alpha )</span>)-homogeneous solutions with <span>(x_1)</span>-symmetric singular profiles <span>((u,p,rho ) in C^{infty }(overline{mathbb {R}^{2}_{+}}backslash {x_1=0}cup {x_2=0})cap C(overline{mathbb {R}^{2}_{+}}))</span> for <span>(-1&lt;alpha &lt; -1/2)</span> and <span>((u,p,rho )in C^{infty }(overline{mathbb {R}^{2}_{+}}backslash {x_1=0}cup {x_2=0}))</span> for <span>(-1/2leqq alpha &lt;1)</span>.</p>\u0000 </li>\u0000 </ul><p> The (<span>(-alpha )</span>)-homogeneous solutions with continuous profiles <span>((u,p,rho )in C^{infty }(overline{mathbb {R}^{2}_{+}} backslash {x_1=0}cup {x_2=0})cap C(overline{mathbb {R}^{2}_{+}}))</span> for <span>(-1&lt;alpha &lt;-1/2)</span> provide examples of self-similar weak solutions in <span>(mathbb {R}^{2}_{+})</span> for the scaling exponent <span>(alpha approx -0.657)</span>, at which Wang et al. (Phys Rev Lett 130(24):244002, 2023) numerically discovered the existence of backward self-similar solutions with smooth profile functions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147938518","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":"On the Singular Limit of Brinkman’s Law to Darcy’s Law","authors":"Noemi David,&nbsp;Matt Jacobs,&nbsp;Inwon Kim","doi":"10.1007/s00205-026-02201-8","DOIUrl":"10.1007/s00205-026-02201-8","url":null,"abstract":"<div><p>In this paper we study singular limits of congestion-averse growth models, connecting different models describing the effect of congestion. These models arise in particular in the context of tissue growth. The main ingredient of our analysis is a family of energy evolution equations and their dissipation structures, which are novel and of independent interest. This strategy allows us to consider a larger family of pressure laws as well as proving the joint limit, from a compressible Brinkman’s model to the incompressible Darcy’s law, where the latter is a Hele-Shaw type free boundary problem.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147938266","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":"Finite Time Blow-Up for the Hypodissipative Navier Stokes Equations with a Force in (L^1_t C_x^{1,epsilon }cap L^{infty }_{t} L^{2}_{x})","authors":"Diego Córdoba,&nbsp;Luis Martínez-Zoroa,&nbsp;Fan Zheng","doi":"10.1007/s00205-026-02198-0","DOIUrl":"10.1007/s00205-026-02198-0","url":null,"abstract":"<div><p>In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by <span>(|nabla |^{alpha })</span> for any <span>(alpha in [0, alpha _0))</span> (<span>(alpha _0 = frac{22-8sqrt{7}}{9} &gt; 0)</span>). We construct solutions in <span>(mathbb {R}^3times [0,T])</span> with a finite <span>(T&gt;0)</span> and with an external forcing which is in <span>(L^1_t([0, T]) C_x^{1,epsilon }cap L^{infty }_{t} L^{2}_{x})</span>, such that for each time <span>(t in [0, T))</span>, the velocity <i>u</i> is in the space <span>(C^infty cap L^2)</span> and such that as the time <i>t</i> approaches the blow-up moment <i>T</i>, the integral <span>(int _0^t |nabla u| text {d}s)</span> tends to infinity. Since this result lays in a well-posedness class, the blow-up is generated by the dynamics of the equation and not by the force itself. This is the first blow-up result for hypodissipative Navier–Stokes in a well-posedness class.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13161257/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147935740","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":"Discretely Self-similar Exterior-Naked Singularities for the Einstein-Scalar Field System","authors":"Serban Cicortas,&nbsp;Christoph Kehle","doi":"10.1007/s00205-026-02194-4","DOIUrl":"10.1007/s00205-026-02194-4","url":null,"abstract":"<div><p>Naked singularity spacetimes in general relativity can be naturally divided into two regions: (i) the region exterior to the past light cone of the singularity, that extends all the way to (an incomplete) future null infinity and ensures the nakedness property (what we will call <i>exterior-naked singularity regions</i>), (ii) the interior fill-in that ensures that the singularity arises from regular initial data. The problem of constructing naked singularities has been resolved for the spherically symmetric Einstein-scalar field system by Christodoulou (Ann Math(2) 140(3):607–653, 1994), but his construction, based on a continuously self-similar ansatz, requires that both the exterior and the interior regions are mildly irregular on the past cone of the singularity. On the other hand, numerical works suggest that there exist naked singularity spacetimes with discrete self-similarity arising from smooth initial data. In this paper, we construct <i>exterior-naked singularity regions</i> with discretely self-similar profiles which are smooth on the past cone of the singularity. We show that the scalar field remains uniformly bounded, but the singularity is characterized by the infinite oscillations of the scalar field and the mass aspect ratio. Our examples require however that the mass aspect ratio is uniformly small, and thus the solutions are distinct from the exterior regions of the numerical examples.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147937856","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":"Gradient Integrability for Bounded (textrm{BD})-Minimizers","authors":"Lisa Beck,&nbsp;Ferdinand Eitler,&nbsp;Franz Gmeineder","doi":"10.1007/s00205-026-02170-y","DOIUrl":"10.1007/s00205-026-02170-y","url":null,"abstract":"<div><p>We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on <span>(textrm{BD}(Omega ))</span> have weak gradients in <span>(textrm{L}_{textrm{loc}}^{1}(Omega ;mathbb {R}^{ntimes n}))</span>. This is achieved for the sharp ellipticity range that is presently known to yield <span>(textrm{W}_{textrm{loc}}^{1,1})</span>-regularity in the full gradient case on <span>(textrm{BV}(Omega ;mathbb {R}^{n}))</span>. As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on <span>(textrm{BD}(Omega ))</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02170-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147937857","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":"Global Existence and Asymptotic Equivalence to Barenblatt-Type Solutions for the Physical Vacuum Free Boundary Problem of Damped Compressible Euler Equations in M-D","authors":"Huihui Zeng","doi":"10.1007/s00205-026-02206-3","DOIUrl":"10.1007/s00205-026-02206-3","url":null,"abstract":"<div><p>For the physical vacuum free boundary problem of the damped compressible Euler equations in both 2D and 3D, we prove the global existence of smooth solutions and justify their time-asymptotic equivalence to the corresponding Barenblatt self-similar solutions derived from the porous media equation under a Darcy’s law approximation, provided that the initial data are small perturbations of the Barenblatt solutions. Building on the 3D almost global existence result in [Zeng, Arch. Ration. Mech. Anal. 239, 553–597 (2021)], our key contribution lies in improving the decay rate of the time derivative of the perturbation from <span>(-1)</span> (as previously established) to <span>(-1-varepsilon )</span> for a fixed constant <span>(varepsilon &gt; 0)</span>. This critical enhancement ensures time integrability and hence global existence. Together with the previous 1D result in [Luo–Zeng, Comm. Pure Appl. Math. 69, 1354–1396 (2016)], the results obtained in this paper provide a complete answer to the question raised in [Liu, T.-P.: Jpn. J. Appl. Math. 13, 25–32 (1996)]. Moreover, we also consider the problem with time-dependent damping of the form <span>((1+t)^{-lambda })</span> for <span>(0&lt; lambda &lt; 1)</span>. Notably, our framework unifies the treatment of both time-dependent (<span>(0&lt; lambda &lt; 1)</span>) and time-independent (<span>(lambda = 0)</span>) damping cases across dimensions. We further quantify the decay rates of the density and velocity, as well as the expansion rate of the physical vacuum boundary.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147938115","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":"Concentration of Cones in the Alt-Phillips Problem","authors":"Ovidiu Savin,&nbsp;Hui Yu","doi":"10.1007/s00205-026-02204-5","DOIUrl":"10.1007/s00205-026-02204-5","url":null,"abstract":"<div><p>We study minimizing cones in the Alt-Phillips problem for when the exponent <span>(gamma )</span> is for close to 1. For when <span>(gamma )</span> converges to 1, we show that the cones concentrate around <i>symmetric</i> solutions to the classical obstacle problem. To be precise, the limiting profiles are radial in a subspace and invariant in directions perpendicular to that subspace.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147830166","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":"On the Existence of Source-Solutions to the Multidimensional Burgers Equation","authors":"João Fernando Nariyoshi","doi":"10.1007/s00205-026-02199-z","DOIUrl":"10.1007/s00205-026-02199-z","url":null,"abstract":"<div><p>We establish that, in general, there are no entropy solutions to the Cauchy problem for the multidimensional Burgers equation when the initial data is a measure. This answers in the negative a conjecture of <span>D. Serre–L. Silvestre</span> (Arch Rat Mech Anal 234:1391–1411, 2019). This conjecture was motivated by the description of the asymptotic behavior of entropy solutions with integrable initial data. Despite this negative result, we still derive some information on the asymptotic behavior of such solutions. Our method relies on new Laplace transform estimates for the propagation of information.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02199-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147829340","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":"Separation of Time Scales in Weakly Interacting Diffusions","authors":"Zachary P. Adams,&nbsp;Maximilian Engel,&nbsp;Rishabh S. Gvalani","doi":"10.1007/s00205-026-02180-w","DOIUrl":"10.1007/s00205-026-02180-w","url":null,"abstract":"<div><p>We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a “droplet state” which is <i>metastable</i>, i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to 0. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is <i>O</i>(1) as the inverse temperature <span>(beta rightarrow infty )</span>. Meanwhile the rate of leakage away from its centre of mass is <span>(O(e^{-beta }))</span>. Furthermore, the quasi-stationary distribution is localised on a length scale of order <span>(O(beta ^{-frac{1}{2}}))</span>. Our proofs rely on understanding the large <span>(beta )</span>-asymptotics of the first two eigenvalues of the generator, which we study using techniques from semiclassical analysis. We thus provide a partial answer to a question posed by Carrillo et al. (see Aggregation–diffusion equations: dynamics, asymptotics, and singular limits. Active particles. Advances in theory, models, and applications, modeling and simulation in science, engineering and technology, vol 2, pp 65–108, Birkhäuser/Springer, Cham, 2019, Section 3.2.2) in the microscopic setting.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02180-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147796957","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":"Long-Time Dynamics for the Kelvin-Helmholtz Equations Close to Circular Vortex Sheets","authors":"Federico Murgante,&nbsp;Emeric Roulley,&nbsp;Stefano Scrobogna","doi":"10.1007/s00205-026-02174-8","DOIUrl":"10.1007/s00205-026-02174-8","url":null,"abstract":"<div><p>We consider the Kelvin-Helmholtz system describing the evolution of a vortex-sheet near the circular stationary solution. Answering previous numerical conjectures in the 1990s physics literature, we prove an almost global existence result for small-amplitude solutions. We first establish the existence of a linear stability threshold for the Weber number, which represents the ratio between the square of the background velocity jump and the surface tension. Then, we prove that, for almost all values of the Weber number below this threshold, any small solution lives for almost all times, remaining close to the equilibrium. Our analysis reveals a remarkable stabilization phenomenon: the presence of both non-zero background velocity jump and capillarity effects enables us to prevent nonlinear instability phenomena, despite the inherently unstable nature of the classical Kelvin-Helmholtz problem. This long-time existence would not be achievable in a setting where capillarity alone provides linear stabilization, without the richer modulation induced by the velocity jump. Our proof exploits the Hamiltonian nature of the equations. More specifically, we employ Hamiltonian Birkhoff normal form techniques for quasi-linear systems together with a general approach for paralinearization of non-linear singular integral operators. This approach allows us to control resonances and quasi-resonances at arbitrary order, ensuring the desired long-time stability result.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13121218/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147789960","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}