Pub Date : 2025-11-30DOI: 10.1016/j.na.2025.114027
André de Laire, Erwan Le Quiniou
We study a defocusing quasilinear Schrödinger equation with nonzero conditions at infinity in dimension one. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross–Pitaevskii equation, and can also be derived by considering the effects of surface tension in superfluids. When the quasilinear term is neglected, the resulting equation is the classical Gross–Pitaevskii equation, which possesses a well-known stable branch of subsonic traveling waves solution, given by dark solitons.
Our goal is to investigate how the quasilinear term and the intensity-dependent dispersion affect the traveling-wave solutions. We provide a complete classification of finite energy traveling waves of the equation, in terms of two parameters: the speed and the strength of the quasilinear term. This classification leads to the existence of dark and antidark solitons, as well as more exotic localized solutions like dark cuspons, compactons, and composite waves, even for supersonic speeds. Depending on the parameters, these types of solutions can coexist, showing that finite energy solutions are not unique. Furthermore, we prove that some of these dark solitons can be obtained as minimizers of the energy, at fixed momentum, and that they are orbitally stable.
{"title":"Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background","authors":"André de Laire, Erwan Le Quiniou","doi":"10.1016/j.na.2025.114027","DOIUrl":"10.1016/j.na.2025.114027","url":null,"abstract":"<div><div>We study a defocusing quasilinear Schrödinger equation with nonzero conditions at infinity in dimension one. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross–Pitaevskii equation, and can also be derived by considering the effects of surface tension in superfluids. When the quasilinear term is neglected, the resulting equation is the classical Gross–Pitaevskii equation, which possesses a well-known stable branch of subsonic traveling waves solution, given by dark solitons.</div><div>Our goal is to investigate how the quasilinear term and the intensity-dependent dispersion affect the traveling-wave solutions. We provide a complete classification of finite energy traveling waves of the equation, in terms of two parameters: the speed and the strength of the quasilinear term. This classification leads to the existence of dark and antidark solitons, as well as more exotic localized solutions like dark cuspons, compactons, and composite waves, even for supersonic speeds. Depending on the parameters, these types of solutions can coexist, showing that finite energy solutions are not unique. Furthermore, we prove that some of these dark solitons can be obtained as minimizers of the energy, at fixed momentum, and that they are orbitally stable.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114027"},"PeriodicalIF":1.3,"publicationDate":"2025-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685496","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 : 2025-11-24DOI: 10.1016/j.na.2025.114015
Bowen Zheng , Tohru Ozawa
<div><div>This paper is dedicated to the blow-up solution for the divergence Schrödinger equations with inhomogeneous nonlinearity (dINLS for short) <span><span><span><math><mrow><mi>i</mi><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>b</mi></mrow></msup><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>c</mi></mrow></msup><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mn>2</mn><mo>−</mo><mi>n</mi><mo><</mo><mi>b</mi><mo><</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>c</mi><mo>></mo><mi>b</mi><mo>−</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mi>p</mi><mo>−</mo><mn>2</mn><mi>c</mi><mo><</mo><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. First, for radial blow-up solutions in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></math></span>, we prove an upper bound on the blow-up rate for the intercritical dNLS. Moreover, an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm concentration in the mass-critical case is also obtained by giving a compact lemma. Next, we turn to the non-radial case. By establishing two types of Gagliardo–Nirenberg inequalities, we show the existence of finite time blow-up solutions in <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>∩</mo><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></mrow></math></span>, where <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup
{"title":"The blow-up dynamics for the divergence Schrödinger equations with inhomogeneous nonlinearity","authors":"Bowen Zheng , Tohru Ozawa","doi":"10.1016/j.na.2025.114015","DOIUrl":"10.1016/j.na.2025.114015","url":null,"abstract":"<div><div>This paper is dedicated to the blow-up solution for the divergence Schrödinger equations with inhomogeneous nonlinearity (dINLS for short) <span><span><span><math><mrow><mi>i</mi><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>b</mi></mrow></msup><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>c</mi></mrow></msup><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mn>2</mn><mo>−</mo><mi>n</mi><mo><</mo><mi>b</mi><mo><</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>c</mi><mo>></mo><mi>b</mi><mo>−</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mi>p</mi><mo>−</mo><mn>2</mn><mi>c</mi><mo><</mo><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>b</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. First, for radial blow-up solutions in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></math></span>, we prove an upper bound on the blow-up rate for the intercritical dNLS. Moreover, an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm concentration in the mass-critical case is also obtained by giving a compact lemma. Next, we turn to the non-radial case. By establishing two types of Gagliardo–Nirenberg inequalities, we show the existence of finite time blow-up solutions in <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>∩</mo><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup></mrow></math></span>, where <span><math><mrow><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>b</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114015"},"PeriodicalIF":1.3,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618595","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 : 2025-11-20DOI: 10.1016/j.na.2025.114018
Irene De Blasi
This work presents some results regarding three-dimensional billiards having a non-constant potential of Keplerian type inside a regular domain . Two models will be analysed: in the first one, only an inner Keplerian potential is present, and every time the particle encounters the boundary of is reflected back by keeping constant its tangential component to , while the normal one changes its sign. The second model is a refractive billiard, where the inner Keplerian potential is coupled with a harmonic outer one; in this case, the interaction with results in a generalised refraction Snell’s law. In both cases, the analysis of a particular type of straight equilibrium trajectories, called homothetic, is carried on, and their presence is linked to the topological chaoticity of the dynamics for large inner energies.
{"title":"Keplerian billiards in three dimensions: Stability of equilibrium orbits and conditions for chaos","authors":"Irene De Blasi","doi":"10.1016/j.na.2025.114018","DOIUrl":"10.1016/j.na.2025.114018","url":null,"abstract":"<div><div>This work presents some results regarding three-dimensional billiards having a non-constant potential of Keplerian type inside a regular domain <span><math><mrow><mi>D</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>. Two models will be analysed: in the first one, only an inner Keplerian potential is present, and every time the particle encounters the boundary of <span><math><mi>D</mi></math></span> is reflected back by keeping constant its tangential component to <span><math><mrow><mi>∂</mi><mi>D</mi></mrow></math></span>, while the normal one changes its sign. The second model is a refractive billiard, where the inner Keplerian potential is coupled with a harmonic outer one; in this case, the interaction with <span><math><mrow><mi>∂</mi><mi>D</mi></mrow></math></span> results in a generalised refraction Snell’s law. In both cases, the analysis of a particular type of straight equilibrium trajectories, called <em>homothetic</em>, is carried on, and their presence is linked to the topological chaoticity of the dynamics for large inner energies.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114018"},"PeriodicalIF":1.3,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571025","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 : 2025-11-19DOI: 10.1016/j.na.2025.114016
R.D. Ayissi , G. Deugoué , J. Ngandjou Zangue , T. Tachim Medjo
In this paper, we study a feedback optimal control problem for the stochastic nonlocal Cahn–Hilliard–Navier–Stokes model in a two-dimensional bounded domain. The model consists of the stochastic Navier–Stokes equations for the velocity, coupled with a nonlocal Cahn-Hilliard system for the order (phase) parameter. We prove the existence of an optimal feedback control for the stochastic nonlocal Cahn–Hilliard-Navier-Stokes system. Moreover using the Galerkin approximation, we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs.
{"title":"On the existence of optimal and ɛ-optimal controls for the stochastic 2D nonlocal Cahn–Hilliard–Navier–Stokes system","authors":"R.D. Ayissi , G. Deugoué , J. Ngandjou Zangue , T. Tachim Medjo","doi":"10.1016/j.na.2025.114016","DOIUrl":"10.1016/j.na.2025.114016","url":null,"abstract":"<div><div>In this paper, we study a feedback optimal control problem for the stochastic nonlocal Cahn–Hilliard–Navier–Stokes model in a two-dimensional bounded domain. The model consists of the stochastic Navier–Stokes equations for the velocity, coupled with a nonlocal Cahn-Hilliard system for the order (phase) parameter. We prove the existence of an optimal feedback control for the stochastic nonlocal Cahn–Hilliard-Navier-Stokes system. Moreover using the Galerkin approximation, we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114016"},"PeriodicalIF":1.3,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145537288","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 : 2025-11-19DOI: 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}
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