Archive for Rational Mechanics and Analysis最新文献_第4页

The Equality Case in the Substatic Heintze–Karcher Inequality Substatic Heintze-Karcher 不等式中的平等案例

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-11-05 DOI: 10.1007/s00205-024-02022-7

Stefano Borghini, Mattia Fogagnolo, Andrea Pinamonti

引用次数: 0

Regularity and compactness for critical points of degenerate polyconvex energies 退化多凸能临界点的正则性和紧凑性

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-11-04 DOI: 10.1007/s00205-024-02055-y

André Guerra, Riccardo Tione

引用次数: 0

Global Small Analytic Solution of 3-D Anisotropic Navier-Stokes System 三维各向异性纳维-斯托克斯系统的全局小解析解

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-30 DOI: 10.1007/s00205-024-02051-2

Ning Liu, Ping Zhang

引用次数: 0

Navigating the Complex Landscape of Shock Filter Cahn–Hilliard Equation: From Regularized to Entropy Solutions 驾驭冲击滤波卡恩-希利亚德方程的复杂局面：从正则化到熵解

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-28 DOI: 10.1007/s00205-024-02057-w

Darko Mitrovic, Andrej Novak

{"title":"Navigating the Complex Landscape of Shock Filter Cahn–Hilliard Equation: From Regularized to Entropy Solutions","authors":"Darko Mitrovic, Andrej Novak","doi":"10.1007/s00205-024-02057-w","DOIUrl":"10.1007/s00205-024-02057-w","url":null,"abstract":"<div>Image inpainting involves filling in damaged or missing regions of an image by utilizing information from the surrounding areas. In this paper, we investigate a fully nonlinear partial differential equation inspired by the modified Cahn–Hilliard equation. Instead of using standard potentials that depend solely on pixel intensities, we consider morphological image enhancement filters that are based on a variant of the shock filter: <div><div>$$begin{aligned} partial _t u&= Delta left( -nu arctan (Delta u)|nabla u| - mu Delta u right) + lambda (u_0 - u). end{aligned}$$</div></div>This is referred to as the Shock Filter Cahn–Hilliard Equation. The equation is nonlinear with respect to the highest-order derivative, which poses significant mathematical challenges. To address these, we make use of a specific approximation argument, establishing the existence of a family of approximate solutions through the Leray–Schauder fixed point theorem and the Aubin–Lions lemma. In the limit, we obtain a solution strategy wherein we can prove the existence and uniqueness of solutions. Proving the latter involves the Kruzhkov entropy type-admissibility conditions. Additionally, we use a numerical method based on the convexity splitting idea to approximate solutions of the nonlinear partial differential equation and achieve fast inpainting results. To demonstrate the effectiveness of our approach, we apply our method to standard binary images and compare it with variations of the Cahn–Hilliard equation commonly used in the field.\u0000</div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142519164","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

1-D Isentropic Euler Flows: Self-similar Vacuum Solutions 一维等熵欧拉流：自相似真空解

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-24 DOI: 10.1007/s00205-024-02054-z

Helge Kristian Jenssen

{"title":"1-D Isentropic Euler Flows: Self-similar Vacuum Solutions","authors":"Helge Kristian Jenssen","doi":"10.1007/s00205-024-02054-z","DOIUrl":"10.1007/s00205-024-02054-z","url":null,"abstract":"<div>We consider one-dimensional self-similar solutions to the isentropic Euler system when the initial data are at vacuum to the left of the origin. For (x>0), the initial velocity and sound speed are of the form (u_0(x)=u_+x^{1-lambda }) and (c_0(x)=c_+x^{1-lambda }), for constants (u_+in mathbb {R}), (c_+>0), (lambda in mathbb {R}). We analyze the resulting solutions in terms of the similarity parameter (lambda ), the adiabatic exponent (gamma ), and the initial (signed) Mach number (text {Ma}=u_+/c_+). Restricting attention to locally bounded data, we find that when the sound speed initially decays to zero in a Hölder manner ((0<lambda <1)), the resulting flow is always defined globally. Furthermore, there are three regimes depending on (text {Ma}): for sufficiently large positive (text {Ma})-values, the solution is continuous and the initial Hölder decay is immediately replaced by (C^1)-decay to vacuum along a stationary vacuum interface; for moderate values of (text {Ma}), the solution is again continuous and with an accelerating vacuum interface along which (c^2) decays linearly to zero (i.e., a “physical singularity”); for sufficiently large negative (text {Ma})-values, the solution contains a shock wave emanating from the initial vacuum interface and propagating into the fluid, together with a physical singularity along an accelerating vacuum interface. In contrast, when the sound speed initially decays to zero in a (C^1) manner ((lambda <0)), a global flow exists only for sufficiently large positive values of (text {Ma}). The non-existence of global solutions for smaller (text {Ma})-values is due to rapid growth of the data at infinity and is unrelated to the presence of a vacuum.</div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142518934","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

From Incommensurate Bilayer Heterostructures to Allen–Cahn: An Exact Thermodynamic Limit 从不相称双层异质结构到艾伦-卡恩：精确的热力学极限

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-24 DOI: 10.1007/s00205-024-02043-2

Michael Hott, Alexander B. Watson, Mitchell Luskin

引用次数: 0

Uniform (C^{1,alpha })-Regularity for Almost-Minimizers of Some Nonlocal Perturbations of the Perimeter 周长的某些非局部扰动的几乎最小化者的（C^{1,α }）均匀规律性

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-22 DOI: 10.1007/s00205-024-02048-x

M. Goldman, B. Merlet, M. Pegon

{"title":"Uniform (C^{1,alpha })-Regularity for Almost-Minimizers of Some Nonlocal Perturbations of the Perimeter","authors":"M. Goldman, B. Merlet, M. Pegon","doi":"10.1007/s00205-024-02048-x","DOIUrl":"10.1007/s00205-024-02048-x","url":null,"abstract":"<div>In this paper, we establish a (C^{1,alpha })-regularity theorem for almost-minimizers of the functional (mathcal {F}_{varepsilon ,gamma }=P-gamma P_{varepsilon }), where (gamma in (0,1)) and (P_{varepsilon }) is a nonlocal energy converging to the perimeter as (varepsilon ) vanishes. Our theorem provides a criterion for (C^{1,alpha })-regularity at a point of the boundary which is uniform as the parameter (varepsilon ) goes to 0. Since the two terms in the energy are of the same order when (varepsilon ) is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for (varepsilon ) small enough, volume-constrained minimizers of (mathcal {F}_{varepsilon ,gamma }) are balls. For small (varepsilon ), this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel G with sufficiently fast decay at infinity.</div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142519076","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 Spherically Symmetric Solutions of the Multidimensional Full Compressible Navier–Stokes Equations with Large Data 大数据下多维全可压缩纳维-斯托克斯方程的全局球对称解

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-22 DOI: 10.1007/s00205-024-02018-3

Gui-Qiang G. Chen, Yucong Huang, Shengguo Zhu

{"title":"Global Spherically Symmetric Solutions of the Multidimensional Full Compressible Navier–Stokes Equations with Large Data","authors":"Gui-Qiang G. Chen, Yucong Huang, Shengguo Zhu","doi":"10.1007/s00205-024-02018-3","DOIUrl":"10.1007/s00205-024-02018-3","url":null,"abstract":"<div>We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier–Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically symmetric, and away from the vacuum. The solutions obtained here are of global finite total relative-energy including the origin, while cavitation may occur as balls centred at the origin of symmetry for which the interfaces between the fluid and the vacuum must be upper semi-continuous in space-time in the Eulerian coordinates. On any region strictly away from the possible vacuum, the velocity and specific internal energy are Hölder continuous, and the density has a uniform upper bound. To achieve this, our main strategy is to regard the Cauchy problem as the limit of a series of carefully designed initial-boundary value problems that are formulated in finite annular regions. For such approximation problems, we can derive uniform a priori estimates that are independent of both the inner and outer radii of the annuli considered in the spherically symmetric Lagrangian coordinates. The entropy inequality is recovered after taking the limit of the outer radius to infinity by using Mazur’s lemma and the convexity of the entropy function, which is required for the limit of the inner radius tending to zero. Then the global weak solutions of the original problem are attained via careful compactness arguments applied to the approximate solutions in the Eulerian coordinates.</div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02018-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142452895","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

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres 硬球稀薄玻色气体基态能量的上限

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-21 DOI: 10.1007/s00205-024-02049-w

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani, Alessandro Olgiati, Giulio Pasqualetti, Benjamin Schlein

引用次数: 0

Strong (L^2 H^2) Convergence of the JKO Scheme for the Fokker–Planck Equation 福克-普朗克方程的 JKO 方案的强（L^2 H^2）收敛性

IF 2.6 1区数学

Archive for Rational Mechanics and Analysis Pub Date : 2024-10-18 DOI: 10.1007/s00205-024-02037-0

Filippo Santambrogio, Gayrat Toshpulatov

{"title":"Strong (L^2 H^2) Convergence of the JKO Scheme for the Fokker–Planck Equation","authors":"Filippo Santambrogio, Gayrat Toshpulatov","doi":"10.1007/s00205-024-02037-0","DOIUrl":"10.1007/s00205-024-02037-0","url":null,"abstract":"<div>Following a celebrated paper by Jordan, Kinderleherer and Otto, it is possible to discretize in time the Fokker–Planck equation (partial _tvarrho =Delta varrho +nabla cdot (varrho nabla V)) by solving a sequence of iterated variational problems in the Wasserstein space, and the sequence of piecewise constant curves obtained from the scheme is known to converge to the solution of the continuous PDE. This convergence is uniform in time valued in the Wasserstein space and also strong in (L^1) in space-time. We prove in this paper, under some assumptions on the domain (a bounded and smooth convex domain) and on the initial datum (which is supposed to be bounded away from zero and infinity and belong to (W^{1,p}) for an exponent p larger than the dimension), that the convergence is actually strong in (L^2_tH^2_x), hence strongly improving open the previously known results in terms of the order of derivation in space. The technique is based on some inequalities, obtained with optimal transport techniques, that can be proven on the discrete sequence of approximate solutions, and that mimic the corresponding continuous computations.</div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142451106","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