Nonlinear Analysis-Theory Methods & Applications最新文献

{"title":"A Blaschke–Petkantschin formula for linear and affine subspaces with application to intersection probabilities","authors":"","doi":"10.1016/j.na.2024.113672","DOIUrl":"10.1016/j.na.2024.113672","url":null,"abstract":"<div><div>Consider a uniformly distributed random linear subspace <span><math><mi>L</mi></math></span> and a stochastically independent random affine subspace <span><math><mi>E</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, both of fixed dimension. For a natural class of distributions for <span><math><mi>E</mi></math></span> we show that the intersection <span><math><mrow><mi>L</mi><mo>∩</mo><mi>E</mi></mrow></math></span> admits a density with respect to the invariant measure. This density depends only on the distance <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mi>L</mi><mo>∩</mo><mi>E</mi></mrow></math></span> to the origin and is derived explicitly. It can be written as the product of a power of <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span> and a part involving an incomplete beta integral. Choosing <span><math><mi>E</mi></math></span> uniformly among all affine subspaces of fixed dimension hitting the unit ball, we derive an explicit density for the random variable <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span> and study the behavior of the probability that <span><math><mrow><mi>E</mi><mo>∩</mo><mi>L</mi></mrow></math></span> hits the unit ball in high dimensions. Lastly, we show that our result can be extended to the setting where <span><math><mi>E</mi></math></span> is tangent to the unit sphere, in which case we again derive the density for <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span>. Our probabilistic results are derived by means of a new integral–geometric transformation formula of Blaschke–Petkantschin type.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global low regularity solutions to the Benjamin equation in weighted spaces","authors":"","doi":"10.1016/j.na.2024.113674","DOIUrl":"10.1016/j.na.2024.113674","url":null,"abstract":"<div><div>We show that the Benjamin equation is globally well-posed for real-valued data in the weighted space <span><span><span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>∩</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>s</mi><mo>−</mo><mn>2</mn><mi>r</mi></mrow></msubsup><mo>≔</mo><mrow><mo>{</mo><mrow><mi>u</mi><mspace></mspace><mo>|</mo><mspace></mspace><msub><mrow><mo>‖</mo><mi>u</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><mover><mrow><mi>u</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>ξ</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>,</mo><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mi>s</mi><mo>−</mo><mn>2</mn><mi>r</mi><mo>)</mo></mrow></mrow></msup><mi>d</mi><mi>ξ</mi><mo>)</mo></mrow></mrow></msub><mo>&lt;</mo><mi>∞</mi></mrow><mo>}</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mn>0</mn><mo>≤</mo><mi>r</mi></mrow></math></span> and <span><math><mrow><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mi>r</mi><mo>&lt;</mo><mi>s</mi></mrow></math></span>. The proof is based on direct extensions of standard linear and bilinear estimates originated in Kenig et al. (1993), Kenig et al. (1996), Linares (1999), Kozono et al. (2001), Colliander et al. (2003), Li and Wu (2010) to the weighted settings.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analytical solutions to the free boundary problem of a two-phase model with radial and cylindrical symmetry","authors":"","doi":"10.1016/j.na.2024.113670","DOIUrl":"10.1016/j.na.2024.113670","url":null,"abstract":"<div><div>In this paper, we study the free boundary problem of an inviscid two-phase model, where we take the pressure function as <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>ρ</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mi>γ</mi></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>γ</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>α</mi><mo>≥</mo><mn>1</mn></mrow></math></span>) with <span><math><mi>n</mi></math></span> and <span><math><mi>ρ</mi></math></span> being the densities of two phases. First, we construct some self-similar analytical solutions for the <span><math><mi>N</mi></math></span>-dimensional radially symmetric case by using some ansatzs, and investigate the spreading rate of the free boundary by using the method of averaged quantities. Second, we extend the results of the <span><math><mi>N</mi></math></span>-dimensional radially symmetric case to the three-dimensional cylindrically symmetric case. Third, we present some analytical solutions for the three-dimensional cylindrically symmetric model with a Coriolis force. From the analytical solutions constructed in this paper, we find that the Coriolis force can prevent the free boundary from spreading out infinitely.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001895/pdfft?md5=f5c63e293b0091e1cef7e731ee5a5250&pid=1-s2.0-S0362546X24001895-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Thin film equations with nonlinear deterministic and stochastic perturbations","authors":"","doi":"10.1016/j.na.2024.113646","DOIUrl":"10.1016/j.na.2024.113646","url":null,"abstract":"<div><p>In this paper we consider stochastic thin-film equation with nonlinear drift terms, colored Gaussian Stratonovich noise, as well as nonlinear colored Wiener noise. By means of Trotter–Kato-type decomposition into deterministic and stochastic parts, we couple both of these dynamics via a discrete-in-time scheme, and establish its convergence to a non-negative weak martingale solution.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001652/pdfft?md5=f3d21fbe23f0caa335b0a9f697a81c70&pid=1-s2.0-S0362546X24001652-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 boundary blow-up problem for real (n−1) Monge–Ampère equation","authors":"","doi":"10.1016/j.na.2024.113669","DOIUrl":"10.1016/j.na.2024.113669","url":null,"abstract":"<div><p>In this paper, we establish a necessary and sufficient condition for the solvability of the real <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> Monge–Ampère equation <span><math><mrow><mover><mrow><mo>det</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></mover><mrow><mo>(</mo><mi>Δ</mi><mi>u</mi><mi>I</mi><mo>−</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> in bounded domains with infinite Dirichlet boundary condition. The <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> Monge–Ampère operator is derived from geometry and has recently received much attention. Our result embraces the case <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>h</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span> is positive and <span><math><mi>f</mi></math></span> satisfies the Keller–Osserman type condition. We describe the asymptotic behavior of the solution by constructing suitable sub-solutions and super-solutions, and obtain a uniqueness result in star-shaped domains by using a scaling technique.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001883/pdfft?md5=dcb2b703c48c88a6c661fc63e5774351&pid=1-s2.0-S0362546X24001883-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ohta–Kawasaki energy for amphiphiles: Asymptotics and phase-field simulations","authors":"","doi":"10.1016/j.na.2024.113665","DOIUrl":"10.1016/j.na.2024.113665","url":null,"abstract":"<div><p>We study the minimizers of a degenerate case of the Ohta–Kawasaki energy, defined as the sum of the perimeter and a Coulombic nonlocal term. We start by investigating radially symmetric candidates which give us insights into the asymptotic behaviors of energy minimizers in the large mass limit. In order to numerically study the problems that are analytically challenging, we propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model. Our phase-field simulations and asymptotic results suggest that the energy minimizers exhibit behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001846/pdfft?md5=c1e8f89875e4f78c3319ad9c2f245a48&pid=1-s2.0-S0362546X24001846-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142171934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large global solutions to the three dimensional compressible flow of liquid crystals","authors":"","doi":"10.1016/j.na.2024.113657","DOIUrl":"10.1016/j.na.2024.113657","url":null,"abstract":"<div><p>The purpose of this paper is to provide a class of large initial data which generates global solutions of the compressible flow of liquid crystals in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. This class of data relax the smallness restriction imposed on the initial incompressible velocity. Moreover, the result improve considerably the work by Hu and Wu [SIAM J. Math. Anal., 45 (2013), 2678-2699].</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001767/pdfft?md5=7de785471e6e5046dbf64fd8ee14f840&pid=1-s2.0-S0362546X24001767-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142171936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation","authors":"","doi":"10.1016/j.na.2024.113658","DOIUrl":"10.1016/j.na.2024.113658","url":null,"abstract":"<div><p>In this paper, we consider the 1D Euler equation with time and space dependent damping term <span><math><mrow><mo>−</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mi>v</mi></mrow></math></span>. It has long been known that when <span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient <span><math><mi>a</mi></math></span> satisfies the following condition <span><span><span><math><mrow><mrow><mo>|</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>|</mo></mrow><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are integrable functions with <span><math><mi>t</mi></math></span> and <span><math><mi>x</mi></math></span>. Under this condition, we show the global existence and the blow-up with small initial data, when <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span> respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001779/pdfft?md5=9f5946837a904defdc71f1e5354348c9&pid=1-s2.0-S0362546X24001779-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142171935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 persistence properties for the fractionary BBM equation with low dispersion in weighted Sobolev spaces","authors":"","doi":"10.1016/j.na.2024.113653","DOIUrl":"10.1016/j.na.2024.113653","url":null,"abstract":"<div><p>We consider the initial value problem associated to the low dispersion fractionary Benjamin–Bona–Mahony equation, fBBM. Our aim is to establish local persistence results in weighted Sobolev spaces and to obtain unique continuation results that imply that those results above are sharp. Hence, arbitrary polynomial type decay is not preserved by the fBBM flow.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X2400172X/pdfft?md5=deedf93280597ef7b37c6bea9b954b83&pid=1-s2.0-S0362546X2400172X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142137191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability of the logarithmic Sobolev inequality and uncertainty principle for the Tsallis entropy","authors":"","doi":"10.1016/j.na.2024.113644","DOIUrl":"10.1016/j.na.2024.113644","url":null,"abstract":"<div><p>We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001639/pdfft?md5=6bfcdd2737c232c0665680eef2bf811d&pid=1-s2.0-S0362546X24001639-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142094884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}