Nonlinear Analysis-Theory Methods & Applications最新文献

{"title":"Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian","authors":"Gyula Csató ,&nbsp;Albert Mas","doi":"10.1016/j.na.2025.113755","DOIUrl":"10.1016/j.na.2025.113755","url":null,"abstract":"<div><div>We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian <span><math><mrow><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities <span><math><mi>f</mi></math></span> in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span> and when <span><math><mrow><mn>2</mn><mi>s</mi><mo>+</mo><mi>β</mi><mo>&lt;</mo><mn>1</mn></mrow></math></span>, the solution is not always <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi><mo>+</mo><mi>β</mi><mo>−</mo><mi>ϵ</mi></mrow></msup></math></span> for all <span><math><mrow><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. Instead, in general the solution <span><math><mi>u</mi></math></span> is at most <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mo>)</mo></mrow></mrow></msup><mo>.</mo></mrow></math></span></div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113755"},"PeriodicalIF":1.3,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163862","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":"Nonlinear stability of viscous shock profiles for a hyperbolic system with Cattaneo’s law and non-convex flux","authors":"Junyuan Deng ,&nbsp;Lan Zhang","doi":"10.1016/j.na.2025.113749","DOIUrl":"10.1016/j.na.2025.113749","url":null,"abstract":"<div><div>This paper is concerned with the large time behavior of solutions to the scalar conservation law with an artificial heat flux term. The heat flux is governed by Cattaneo’s law, which leads to a 2 × 2 system of hyperbolic equations. The existence and nonlinear stability of rarefaction waves and viscous shock waves have been derived under the assumption that flux function is strictly convex. In the current paper, we focus on the one-dimensional Cauchy problem for the system which allows for non-convex flux. Under Oleinik entropy condition, we obtain the existence and asymptotic stability of shifted viscous shock waves with sufficiently small wave strength. The proof is based on the standard energy method and shift theory.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113749"},"PeriodicalIF":1.3,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136925","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":"Existence and boundedness of weak solutions to some vectorial Dirichlet problems","authors":"G.R. Cirmi ,&nbsp;S. D’Asero ,&nbsp;S. Leonardi ,&nbsp;F. Leonetti ,&nbsp;E. Rocha ,&nbsp;V. Staicu","doi":"10.1016/j.na.2025.113751","DOIUrl":"10.1016/j.na.2025.113751","url":null,"abstract":"<div><div>For integers <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mi>Ω</mi></math></span> a bounded subset of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, we prove existence and boundedness of a weak solution <span><math><mi>u</mi></math></span> of the following prototype of nonlinear vectorial Dirichlet problem <span><span><span><math><mfenced><mrow><mtable><mtr><mtd></mtd><mtd><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>−</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mspace></mspace><msup><mrow><mi>u</mi></mrow><mrow><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow><mo>+</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>ν</mi></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>for any <span><math><mrow><mi>x</mi><mo>∈</mo><mi>Ω</mi></mrow></math></span> and <span><math><mrow><mi>ν</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></math></span>, where <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ν</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>ν</mi></mrow></msup></math></span> denote the <span><math><mrow><mi>ν</mi><mo>−</mo></mrow></math></span>th component of the vectors <span><math><mi>u</mi></math></span> and <span><math><mi>F</mi></math></span>, respectively, and the tensor <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> satisfies suitable structural assumptions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113751"},"PeriodicalIF":1.3,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136923","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":"A regularity result for the Fokker–Planck equation with non-smooth drift and diffusion","authors":"Paolo Bonicatto ,&nbsp;Gennaro Ciampa ,&nbsp;Gianluca Crippa","doi":"10.1016/j.na.2025.113748","DOIUrl":"10.1016/j.na.2025.113748","url":null,"abstract":"<div><div>The goal of this paper is to study weak solutions of the Fokker–Planck equation. We first discuss existence and uniqueness of weak solutions in an irregular context, providing a unified treatment of the available literature along with some extensions. Then, we prove a regularity result for distributional solutions under suitable integrability assumptions, relying on a new, simple commutator estimate in the spirit of DiPerna-Lions’ theory of renormalized solutions for the transport equation. Our result is somehow transverse to Theorem 4.3 of Figally (2008): on the diffusion matrix we relax the assumption of Lipschitz regularity in time at the price of assuming Sobolev regularity in space, and we prove the regularity (and hence the uniqueness) of distributional solutions to the Fokker–Planck equation.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113748"},"PeriodicalIF":1.3,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136924","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":"Measure-valued solutions of scalar hyperbolic conservation laws, Part 2: Uniqueness","authors":"Michiel Bertsch ,&nbsp;Flavia Smarrazzo ,&nbsp;Andrea Terracina ,&nbsp;Alberto Tesei","doi":"10.1016/j.na.2024.113740","DOIUrl":"10.1016/j.na.2024.113740","url":null,"abstract":"<div><div>We prove well-posedness of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension, where the singular part of the initial data is a finite superposition of Dirac masses and the flux is a continuous function with possible linear growth at infinity. The uniqueness class consists of signed Radon measure-valued entropy solutions, called <em>admissible</em>, whose regular and singular parts satisfy so-called compatibility conditions and suitable continuity requirements with respect to time.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113740"},"PeriodicalIF":1.3,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136926","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":"Global solutions of the 3D incompressible inhomogeneous viscoelastic system","authors":"Chengfei Ai ,&nbsp;Yong Wang","doi":"10.1016/j.na.2025.113747","DOIUrl":"10.1016/j.na.2025.113747","url":null,"abstract":"<div><div>In this paper, we prove the global existence of strong solutions for the 3D incompressible inhomogeneous viscoelastic system. We avoid to use the “initial state” assumption and the “div–curl” structure in the proof of global solutions inspired by the works (Zhu,2018; Zhu,2022). It is a key to transform the original system into a suitable dissipative system by introducing a new effective tensor, which is useful to establish a series of energy estimates with appropriate time weights.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113747"},"PeriodicalIF":1.3,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136922","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":"Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective","authors":"Juan Carlos Sampedro","doi":"10.1016/j.na.2025.113746","DOIUrl":"10.1016/j.na.2025.113746","url":null,"abstract":"<div><div>In this paper we use abstract bifurcation theory for Fredholm operators of index zero to deal with periodic even solutions of the one-dimensional equation <span><math><mrow><mi>L</mi><mi>u</mi><mo>=</mo><mi>λ</mi><mi>u</mi><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup></mrow></math></span>, where <span><math><mi>L</mi></math></span> is a nonlocal pseudodifferential operator defined as a Fourier multiplier and <span><math><mi>λ</mi></math></span> is the bifurcation parameter. Our general setting includes the fractional Laplacian <span><math><mrow><mi>L</mi><mo>≡</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup></mrow></math></span> and sharpens the results obtained for this operator to date. As a direct application, we establish the existence of traveling waves for general nonlocal dispersive equations for some velocity ranges.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113746"},"PeriodicalIF":1.3,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136921","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":"Boundary value problems for Choquard equations","authors":"Chiara Bernardini ,&nbsp;Annalisa Cesaroni","doi":"10.1016/j.na.2024.113745","DOIUrl":"10.1016/j.na.2024.113745","url":null,"abstract":"<div><div>We consider the following nonlinear Choquard equation <span><math><mrow><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>V</mi><mi>u</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>∗</mo><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><mtext>in</mtext><mspace></mspace><mspace></mspace><mspace></mspace><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mo>+</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is a continuous radial function such that <span><math><mrow><msub><mrow><mo>inf</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi>Ω</mi></mrow></msub><mi>V</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is the Riesz potential of order <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span>. Assuming Neumann or Dirichlet boundary conditions, we prove existence of a positive radial solution to the corresponding boundary value problem when <span><math><mi>Ω</mi></math></span> is an annulus, or an exterior domain of the form <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>∖</mo><mover><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>¯</mo></mover></mrow></math></span>. We also provide a nonexistence result: if <span><math><mrow><mi>p</mi><mo>≥</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span> the corresponding Dirichlet problem has no nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting <span><math><mrow><mi>α</mi><mo>→</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></math></span> we recover existence results for the corresponding <em>local</em> problem with power-type nonlinearity.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113745"},"PeriodicalIF":1.3,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136810","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":"Mild regularity of weak solutions to the Navier – Stokes equations","authors":"Ira Herbst","doi":"10.1016/j.na.2024.113744","DOIUrl":"10.1016/j.na.2024.113744","url":null,"abstract":"<div><div>We use a well known integral equation to derive some regularity properties of Leray–Hopf weak solutions of the Navier–Stokes equations in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow><mo>,</mo><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113744"},"PeriodicalIF":1.3,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136811","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":"Convergence of the area functional on spaces with lower Ricci bounds and applications","authors":"Alessandro Cucinotta","doi":"10.1016/j.na.2024.113738","DOIUrl":"10.1016/j.na.2024.113738","url":null,"abstract":"<div><div>We show that the heat flow provides good approximation properties for the area functional on proper <span><math><mrow><mi>RCD</mi><mrow><mo>(</mo><mi>K</mi><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> spaces, implying that in this setting the area formula for functions of bounded variation holds and that the area functional coincides with its relaxation. We then obtain partial regularity and uniqueness results for functions whose hypographs are perimeter minimizing. Finally, we consider sequences of <span><math><mrow><mi>RCD</mi><mrow><mo>(</mo><mi>K</mi><mo>,</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span> spaces and we show that, thanks to the previously obtained properties, Sobolev minimizers of the area functional in a limit space can be approximated with minimizers along the converging sequence of spaces. Using this last result, we obtain applications on Ricci-limit spaces.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113738"},"PeriodicalIF":1.3,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136809","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}