Pub Date : 2025-11-19DOI: 10.1016/j.na.2025.114007
Jin Tan
We show the global-in-time existence and uniqueness of solutions to the 3D incompressible Hall-magnetohydrodynamic (Hall-MHD) system with small initial data in critical Sobolev spaces. Our result works for general physical parameters, thus gives a full answer to the problem proposed by Chae and Lee in the Remark 2 of Chae and Lee (2014). Moreover, considering the so-called 2D flows for the Hall-MHD system (that is 3D flows independent of the vertical variable), we show that under the sole assumption that the initial magnetic field is small in the critical Sobolev space leads to a global unique solvability statement. Comparing with the classical MHD system, the new difficulties of proving such results come from the additional Hall term, which endows the magnetic equation with a quasi-linear character.
{"title":"Global Fujita-Kato solutions for the incompressible Hall-MHD system","authors":"Jin Tan","doi":"10.1016/j.na.2025.114007","DOIUrl":"10.1016/j.na.2025.114007","url":null,"abstract":"<div><div>We show the global-in-time existence and uniqueness of solutions to the 3D incompressible Hall-magnetohydrodynamic (Hall-MHD) system with small initial data in critical Sobolev spaces. Our result works for general physical parameters, thus gives a full answer to the problem proposed by Chae and Lee in the Remark 2 of Chae and Lee (2014). Moreover, considering the so-called 2<span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>D flows for the Hall-MHD system (that is 3D flows independent of the vertical variable), we show that under the sole assumption that the initial magnetic field is small in the critical Sobolev space leads to a global unique solvability statement. Comparing with the classical MHD system, the new difficulties of proving such results come from the additional Hall term, which endows the magnetic equation with a quasi-linear character.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114007"},"PeriodicalIF":1.3,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145537289","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}
Pub Date : 2025-11-14DOI: 10.1016/j.na.2025.114017
Junior da S. Bessa , João Vitor da Silva , Maria N.B. Frederico , Gleydson C. Ricarte
<div><div>In this manuscript, we establish global weighted Orlicz-Sobolev and variable exponent Morrey–Sobolev estimates for viscosity solutions to fully nonlinear parabolic equations subject to oblique boundary conditions on a portion of the boundary, within the following framework: <span><math><mfenced><mrow><mtable><mtr><mtd><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd><mo>=</mo></mtd><mtd><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>in</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>β</mi><mi>⋅</mi><mi>D</mi><mi>u</mi><mo>+</mo><mi>γ</mi><mi>u</mi></mtd><mtd><mo>=</mo></mtd><mtd><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span> where <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> denotes the parabolic cylinder with spatial base <span><math><mi>Ω</mi></math></span> (a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) and temporal height <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>∂</mi><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. Additionally, <span><math><mi>f</mi></math></span> represents the source term of the parabolic equation, while the boundary data are given by <span><math><mi>β</mi></math></span>, <span><math><mi>γ</mi></math></span>, and <span><math><mi>g</mi></math></span>. Our first main result is a global weighted Orlicz–Sobolev estimate for the solution, obtained under asymptotic structural conditions on the differential operator and appropriate assumptions on the boundary data, assuming that the source term belongs to the corresponding weighted Orlicz space. Leveraging these estimates, we demonstrate several applications, including a density result
{"title":"Weighted Orlicz-Sobolev and variable exponent Morrey regularity for fully nonlinear parabolic PDEs with oblique boundary conditions and applications","authors":"Junior da S. Bessa , João Vitor da Silva , Maria N.B. Frederico , Gleydson C. Ricarte","doi":"10.1016/j.na.2025.114017","DOIUrl":"10.1016/j.na.2025.114017","url":null,"abstract":"<div><div>In this manuscript, we establish global weighted Orlicz-Sobolev and variable exponent Morrey–Sobolev estimates for viscosity solutions to fully nonlinear parabolic equations subject to oblique boundary conditions on a portion of the boundary, within the following framework: <span><math><mfenced><mrow><mtable><mtr><mtd><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd><mo>=</mo></mtd><mtd><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>in</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>β</mi><mi>⋅</mi><mi>D</mi><mi>u</mi><mo>+</mo><mi>γ</mi><mi>u</mi></mtd><mtd><mo>=</mo></mtd><mtd><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span> where <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> denotes the parabolic cylinder with spatial base <span><math><mi>Ω</mi></math></span> (a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) and temporal height <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>∂</mi><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. Additionally, <span><math><mi>f</mi></math></span> represents the source term of the parabolic equation, while the boundary data are given by <span><math><mi>β</mi></math></span>, <span><math><mi>γ</mi></math></span>, and <span><math><mi>g</mi></math></span>. Our first main result is a global weighted Orlicz–Sobolev estimate for the solution, obtained under asymptotic structural conditions on the differential operator and appropriate assumptions on the boundary data, assuming that the source term belongs to the corresponding weighted Orlicz space. Leveraging these estimates, we demonstrate several applications, including a density result ","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114017"},"PeriodicalIF":1.3,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520546","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}
Pub Date : 2025-11-12DOI: 10.1016/j.na.2025.114006
Alessandra De Luca
In the present paper, which aims at representing an improvement of De Luca and Felli (2021), we prove the validity of the strong unique continuation property for solutions to some second order elliptic equations from the edge of a crack via a description of their local behaviour. In particular we relax the star-shapedness condition on the complement of the crack considered in De Luca and Felli (2021) by applying a suitable diffeomorphism which straightens the boundary of the crack before performing an approximation of the fractured domain needed to derive a monotonicity formula
{"title":"A note on unique continuation from the edge of a crack with no star-shapedness condition","authors":"Alessandra De Luca","doi":"10.1016/j.na.2025.114006","DOIUrl":"10.1016/j.na.2025.114006","url":null,"abstract":"<div><div>In the present paper, which aims at representing an improvement of De Luca and Felli (2021), we prove the validity of the strong unique continuation property for solutions to some second order elliptic equations from the edge of a crack via a description of their local behaviour. In particular we relax the star-shapedness condition on the complement of the crack considered in De Luca and Felli (2021) by applying a suitable diffeomorphism which straightens the boundary of the crack before performing an approximation of the fractured domain needed to derive a monotonicity formula</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114006"},"PeriodicalIF":1.3,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520547","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}
Pub Date : 2025-11-12DOI: 10.1016/j.na.2025.113981
Jaume Giné , Dmitry I. Sinelshchikov
It has been conjectured that the only mechanisms capable of producing a center – whether degenerate or not – at a singular point of a polynomial differential system are algebraic reducibility and Liouvillian integrability. In this work, we present an example that is algebraically reducible but neither orbitally reversible nor Liouvillian integrable. The construction of this example is based on a recently developed mechanism that establishes a necessary and sufficient condition for the existence of a center.
{"title":"A new mechanism for producing degenerate centers in polynomial differential systems","authors":"Jaume Giné , Dmitry I. Sinelshchikov","doi":"10.1016/j.na.2025.113981","DOIUrl":"10.1016/j.na.2025.113981","url":null,"abstract":"<div><div>It has been conjectured that the only mechanisms capable of producing a center – whether degenerate or not – at a singular point of a polynomial differential system are algebraic reducibility and Liouvillian integrability. In this work, we present an example that is algebraically reducible but neither orbitally reversible nor Liouvillian integrable. The construction of this example is based on a recently developed mechanism that establishes a necessary and sufficient condition for the existence of a center.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113981"},"PeriodicalIF":1.3,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520545","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}
Pub Date : 2025-11-07DOI: 10.1016/j.na.2025.114005
Kush Kinra , Fernanda Cipriano
In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincaré domains (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. Firstly, we prove that the underlying system defined on has a unique weak solution (in the analytic sense) under Dirichlet boundary condition and it also generates random dynamical system . Secondly, we consider the underlying system on bounded domains. Using the compact Sobolev embedding , we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in . Thirdly, we consider the underlying system on unbounded Poincaré domains with external forcing in and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding , we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of . Note that due to the presence of several nonlinear terms in the underlying system, we are not able to use the energy equality method to obtain the asymptotic compactness of in unbounded domains, which makes the analysis of this work in unbounded domains more difficult and interesting. Finally, as a consequence of the existence of random attractors, we address the existence of invariant measures for underlying system. To the best of authors’ knowledge, this is the first work which consider a class of the 2D as well as 3D incompressible stochastic third-grade fluids equations and establish the existence of random attractor in bounded as well as unbounded domains. In addition, this is the first work which address the existence of invariant measures for underlying system on unbounded domains.
{"title":"Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincaré domains","authors":"Kush Kinra , Fernanda Cipriano","doi":"10.1016/j.na.2025.114005","DOIUrl":"10.1016/j.na.2025.114005","url":null,"abstract":"<div><div>In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincaré domains <span><math><mi>O</mi></math></span> (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. Firstly, we prove that the underlying system defined on <span><math><mi>O</mi></math></span> has a unique weak solution (in the analytic sense) under Dirichlet boundary condition and it also generates random dynamical system <span><math><mi>Ψ</mi></math></span>. Secondly, we consider the underlying system on bounded domains. Using the compact Sobolev embedding <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow><mo>↪</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span>, we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span>. Thirdly, we consider the underlying system on unbounded Poincaré domains with external forcing in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span> and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow><mo>↪</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span>, we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of <span><math><mi>Ψ</mi></math></span>. Note that due to the presence of several nonlinear terms in the underlying system, we are not able to use the energy equality method to obtain the asymptotic compactness of <span><math><mi>Ψ</mi></math></span> in unbounded domains, which makes the analysis of this work in unbounded domains more difficult and interesting. Finally, as a consequence of the existence of random attractors, we address the existence of invariant measures for underlying system. To the best of authors’ knowledge, this is the first work which consider a class of the 2D as well as 3D incompressible stochastic third-grade fluids equations and establish the existence of random attractor in bounded as well as unbounded domains. In addition, this is the first work which address the existence of invariant measures for underlying system on unbounded domains.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114005"},"PeriodicalIF":1.3,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467936","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}
Pub Date : 2025-11-06DOI: 10.1016/j.na.2025.113997
Daniel Spector , Dmitriy Stolyarov
In this paper, we define spaces of measures with dimensional stability . These spaces bridge between , the space of finite Radon measures, and , the real Hardy space. We show the spaces support Sobolev inequalities for , while for any we show that the lower Hausdorff dimension of an element of is at least .
{"title":"On dimension stable spaces of measures","authors":"Daniel Spector , Dmitriy Stolyarov","doi":"10.1016/j.na.2025.113997","DOIUrl":"10.1016/j.na.2025.113997","url":null,"abstract":"<div><div>In this paper, we define spaces of measures <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with dimensional stability <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>. These spaces bridge between <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, the space of finite Radon measures, and <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, the real Hardy space. We show the spaces <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> support Sobolev inequalities for <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>]</mo></mrow></mrow></math></span>, while for any <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>]</mo></mrow></mrow></math></span> we show that the lower Hausdorff dimension of an element of <span><math><mrow><mi>D</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> is at least <span><math><mi>β</mi></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113997"},"PeriodicalIF":1.3,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467932","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}
Pub Date : 2025-11-05DOI: 10.1016/j.na.2025.114000
V. Angulo-Castillo , L.C.F. Ferreira , E.J. Villamizar-Roa
This paper addresses the -dimensional elliptic–parabolic Keller–Segel system by employing a Fourier-based framework. This approach allows to consider a new class of -initial data for the associated Cauchy problem, which can be arbitrarily large in norm. More specifically, our main result establishes global-in-time well-posedness for initial data belonging to a space of tempered distributions whose Fourier transform is a Radon measure supported on sum-closed frequency sets at a distance from the origin. The construction of large solutions can be carried out by taking the distance sufficiently large, a feasible condition that can be handled in practice through suitable translations in Fourier variables. The initial class includes, in particular, data of non-decaying type as as well as periodic and almost periodic functions.
{"title":"Large global bounded solutions for the parabolic–elliptic Keller–Segel system via a framework of sum-closed frequency sets","authors":"V. Angulo-Castillo , L.C.F. Ferreira , E.J. Villamizar-Roa","doi":"10.1016/j.na.2025.114000","DOIUrl":"10.1016/j.na.2025.114000","url":null,"abstract":"<div><div>This paper addresses the <span><math><mi>n</mi></math></span>-dimensional elliptic–parabolic Keller–Segel system by employing a Fourier-based framework. This approach allows to consider a new class of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-initial data for the associated Cauchy problem, which can be arbitrarily large in norm. More specifically, our main result establishes global-in-time well-posedness for initial data belonging to a space of tempered distributions whose Fourier transform is a Radon measure supported on sum-closed frequency sets at a distance <span><math><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math></span> from the origin. The construction of large solutions can be carried out by taking the distance <span><math><mi>δ</mi></math></span> sufficiently large, a feasible condition that can be handled in practice through suitable translations in Fourier variables. The initial class includes, in particular, data of non-decaying type as <span><math><mrow><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mi>∞</mi><mo>,</mo></mrow></math></span> as well as periodic and almost periodic functions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114000"},"PeriodicalIF":1.3,"publicationDate":"2025-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467930","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}
Pub Date : 2025-11-05DOI: 10.1016/j.na.2025.114004
David Jesus , María Soria-Carro
We consider transmission problems for parabolic equations governed by distinct fully nonlinear operators on each side of a time-dependent interface. We prove that if the interface is , in the parabolic sense, then viscosity solutions are piecewise up to the interface. As byproducts, we obtain a new ABP–Krylov–Tso estimate, and establish existence, uniqueness, a comparison principle, and regularity results for the flat interface problem.
{"title":"Fully nonlinear parabolic fixed transmission problems","authors":"David Jesus , María Soria-Carro","doi":"10.1016/j.na.2025.114004","DOIUrl":"10.1016/j.na.2025.114004","url":null,"abstract":"<div><div>We consider transmission problems for parabolic equations governed by distinct fully nonlinear operators on each side of a time-dependent interface. We prove that if the interface is <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>, in the parabolic sense, then viscosity solutions are piecewise <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> up to the interface. As byproducts, we obtain a new ABP–Krylov–Tso estimate, and establish existence, uniqueness, a comparison principle, and regularity results for the flat interface problem.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114004"},"PeriodicalIF":1.3,"publicationDate":"2025-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467931","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}
Pub Date : 2025-11-05DOI: 10.1016/j.na.2025.113998
Stefano Almi , Dario Reggiani , Francesco Solombrino
We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity with dislocations. Precisely, we show that the -distance of a possibly incompatible strain field from a single well is controlled in terms of the -distance from a finite set of wells, of , and of . As a consequence, we derive a strain-gradient plasticity model as -limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of Alicandro et al. (2018) to the case of incompatible vector fields.
{"title":"Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity","authors":"Stefano Almi , Dario Reggiani , Francesco Solombrino","doi":"10.1016/j.na.2025.113998","DOIUrl":"10.1016/j.na.2025.113998","url":null,"abstract":"<div><div>We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity with dislocations. Precisely, we show that the <span><math><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>∗</mo></mrow></msup></mrow></msup></math></span>-distance of a possibly incompatible strain field <span><math><mi>β</mi></math></span> from a single well is controlled in terms of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>∗</mo></mrow></msup></mrow></msup></math></span>-distance from a finite set of wells, of <span><math><mrow><mi>curl</mi><mi>β</mi></mrow></math></span>, and of <span><math><mrow><mi>div</mi><mi>β</mi></mrow></math></span>. As a consequence, we derive a strain-gradient plasticity model as <span><math><mi>Γ</mi></math></span>-limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of Alicandro et al. (2018) to the case of incompatible vector fields.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113998"},"PeriodicalIF":1.3,"publicationDate":"2025-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467934","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}
Pub Date : 2025-11-05DOI: 10.1016/j.na.2025.114003
Willian Cintra , Cristian Morales-Rodrigo , Antonio Suárez
In this work, we analyze a stationary degenerate logistic equation with both local and non-local diffusion. Primarily employing bifurcation results, sub- and supersolution methods, and maximum principles, we establish results regarding the existence, non-existence, and uniqueness of positive solutions. Additionally, using appropriate large solutions, we conduct a detailed study of the asymptotic behavior of the solutions with respect to one of the equation’s parameters, showing that the presence of the non-local diffusion can drastically change this pointwise behavior when compared with the local case.
{"title":"Asymptotic behavior of positive solutions for a degenerate logistic equation with mixed local and non-local diffusion","authors":"Willian Cintra , Cristian Morales-Rodrigo , Antonio Suárez","doi":"10.1016/j.na.2025.114003","DOIUrl":"10.1016/j.na.2025.114003","url":null,"abstract":"<div><div>In this work, we analyze a stationary degenerate logistic equation with both local and non-local diffusion. Primarily employing bifurcation results, sub- and supersolution methods, and maximum principles, we establish results regarding the existence, non-existence, and uniqueness of positive solutions. Additionally, using appropriate large solutions, we conduct a detailed study of the asymptotic behavior of the solutions with respect to one of the equation’s parameters, showing that the presence of the non-local diffusion can drastically change this pointwise behavior when compared with the local case.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114003"},"PeriodicalIF":1.3,"publicationDate":"2025-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467933","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}
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