Pub Date : 2026-01-28DOI: 10.1016/j.na.2026.114065
Jie Li
<div><div>In this paper, we consider the cauchy problem for the stochastic regularized dispersive wave (SDW) equations forced by the Gaussian process<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>L</mi><mo>)</mo></mrow><mi>d</mi><mi>u</mi><mo>+</mo><mrow><mo>(</mo><msub><mi>u</mi><mi>x</mi></msub><mo>+</mo><msub><mrow><mo>(</mo><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>x</mi></msub><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>=</mo><mstyle><mi>Φ</mi></mstyle><mi>d</mi><mi>W</mi><mo>,</mo><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow><mspace></mspace><mspace></mspace><mspace></mspace><mrow><mo>(</mo><mn>0.1</mn><mo>)</mo></mrow></mrow></math></span></span></span>where <span><math><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is a real-valued function and <em>W</em> is a two-parameter Gaussian white noise on <span><math><mrow><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup></mrow></math></span>. <em>L</em> is a Fourier multiplier operator and has a real representation <em>θ</em>(<em>ξ</em>) under the Fourier action. <em>h</em> is a real-valued, smooth function of one real variable. Φ is a Hilbert-Schmidt operator. Local well-posedness of (0.1) is obtained for <em>H<sup>s</sup></em> initial data, almost surely. If <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow><mi>r</mi></msup></mrow></math></span> with <em>r</em> > 1, <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> and <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>u</mi><mn>2</mn></msup></mrow></math></span>, global well-posedness of (0.1)is obtained for <em>H<sup>s</sup></em> initial data, almost surely. Moreover, this essay also shows this global solution <span><math><mrow><mi>u</mi><mo>∈</mo><msubsup><mi>L</mi><mi>F</mi><msup><mn>2</mn><mi>α</mi></msup></msubsup><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>;</mo><mi>C</mi><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>T</mi><mn>0</mn></msub><mo>]</mo></mrow><mo>;</mo><msup><mi>H</mi><mi>s</mi></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> for any <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> (<em>r</em> > 1) and any <span><math><mrow><mi>α</mi><mo>∈</mo><msup><mi>Z</mi><mo>+</mo></msup></mrow></math></s
本文考虑高斯过程{(I+L)du+(ux+(h(u))x)dt=ΦdW,(x,t)∈R×R+,u(x,0)=u0,x∈R,（0.1）所迫随机正则化色散波（SDW）方程的柯西问题，其中u=u（x,t）是实值函数，W是R×R+上的双参数高斯白噪声。L是傅里叶乘数算子在傅里叶作用下有一个实数表示θ（ξ）H是一个单实变量的实值光滑函数。Φ是Hilbert-Schmidt算子。初始数据的局部适定性为（0.1），几乎可以肯定。如果θ(ξ)=|ξ|r, r >； 1,s≥r2, h(u)=12u2，则Hs初始数据的全局适定性为（0.1），几乎可以肯定。此外，本文还给出了对于任意s≥r2 （R >； 1）和任意α∈Z+的全局解u∈LF2α（Ω；C([0,T0];Hs(R))）。
{"title":"Cauchy problem for stochastic regularized nonlinear dispersive wave equations","authors":"Jie Li","doi":"10.1016/j.na.2026.114065","DOIUrl":"10.1016/j.na.2026.114065","url":null,"abstract":"<div><div>In this paper, we consider the cauchy problem for the stochastic regularized dispersive wave (SDW) equations forced by the Gaussian process<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>L</mi><mo>)</mo></mrow><mi>d</mi><mi>u</mi><mo>+</mo><mrow><mo>(</mo><msub><mi>u</mi><mi>x</mi></msub><mo>+</mo><msub><mrow><mo>(</mo><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>x</mi></msub><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>=</mo><mstyle><mi>Φ</mi></mstyle><mi>d</mi><mi>W</mi><mo>,</mo><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow><mspace></mspace><mspace></mspace><mspace></mspace><mrow><mo>(</mo><mn>0.1</mn><mo>)</mo></mrow></mrow></math></span></span></span>where <span><math><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is a real-valued function and <em>W</em> is a two-parameter Gaussian white noise on <span><math><mrow><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup></mrow></math></span>. <em>L</em> is a Fourier multiplier operator and has a real representation <em>θ</em>(<em>ξ</em>) under the Fourier action. <em>h</em> is a real-valued, smooth function of one real variable. Φ is a Hilbert-Schmidt operator. Local well-posedness of (0.1) is obtained for <em>H<sup>s</sup></em> initial data, almost surely. If <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow><mi>r</mi></msup></mrow></math></span> with <em>r</em> > 1, <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> and <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>u</mi><mn>2</mn></msup></mrow></math></span>, global well-posedness of (0.1)is obtained for <em>H<sup>s</sup></em> initial data, almost surely. Moreover, this essay also shows this global solution <span><math><mrow><mi>u</mi><mo>∈</mo><msubsup><mi>L</mi><mi>F</mi><msup><mn>2</mn><mi>α</mi></msup></msubsup><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>;</mo><mi>C</mi><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>T</mi><mn>0</mn></msub><mo>]</mo></mrow><mo>;</mo><msup><mi>H</mi><mi>s</mi></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> for any <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> (<em>r</em> > 1) and any <span><math><mrow><mi>α</mi><mo>∈</mo><msup><mi>Z</mi><mo>+</mo></msup></mrow></math></s","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114065"},"PeriodicalIF":1.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078440","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 : 2026-01-27DOI: 10.1016/j.na.2026.114064
Caihong Chang , Zhengce Zhang
In this paper we prove two properties of positive weak solutions for quasilinear elliptic equation of the type , with f satisfying certain structure conditions and involving the product of the function and its gradient. First, we establish a priori estimates for all solutions by utilizing the well-known doubling lemma. Then, we use topological degree to prove the existence of positive weak solutions. Our proof is based on a priori bounds, which will be achieved by applying a blow-up technique developed in [Rev. Mat. Iberoam. 34 (2018) 195–220]. Since the gradient of solution is singular near the boundary, we adopt a suitable weighted norm that involves the distance function to describe this singularity, and then add the restrictions on the exponents of quasilinear equations to the exponent of the weight terms, thereby extending the assumptions regarding upper bounds on exponent of solution from Serrin exponent presented in [Nonlinear Anal. 220 (2020) 112873] to Sobolev exponent.
{"title":"A priori estimates and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms","authors":"Caihong Chang , Zhengce Zhang","doi":"10.1016/j.na.2026.114064","DOIUrl":"10.1016/j.na.2026.114064","url":null,"abstract":"<div><div>In this paper we prove two properties of positive weak solutions for quasilinear elliptic equation of the type <span><math><mrow><mo>−</mo><msub><mstyle><mi>Δ</mi></mstyle><mi>m</mi></msub><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>∇</mi><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, with <em>f</em> satisfying certain structure conditions and involving the product of the function and its gradient. First, we establish a priori estimates for all solutions by utilizing the well-known doubling lemma. Then, we use topological degree to prove the existence of positive weak solutions. Our proof is based on a priori bounds, which will be achieved by applying a blow-up technique developed in [Rev. Mat. Iberoam. 34 (2018) 195–220]. Since the gradient of solution is singular near the boundary, we adopt a suitable weighted norm that involves the distance function to describe this singularity, and then add the restrictions on the exponents of quasilinear equations to the exponent of the weight terms, thereby extending the assumptions regarding upper bounds on exponent of solution from Serrin exponent presented in [Nonlinear Anal. 220 (2020) 112873] to Sobolev exponent.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114064"},"PeriodicalIF":1.3,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078438","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 : 2026-01-27DOI: 10.1016/j.na.2026.114061
Shokhrukh Y. Kholmatov , Paolo Piovano
The existence and the regularity results obtained in [41] for the variational model introduced in [40] to study the optimal shape of crystalline materials in the setting of stress-driven rearrangement instabilities (SDRI) are extended from two dimensions to any dimensions n ≥ 2. The energy is the sum of the elastic and the surface energy contributions, which cannot be decoupled, and depend on configurational pairs consisting of a set and a function that model the region occupied by the crystal and the bulk displacement field, respectively. By following the physical literature, the “driving stress” due to the mismatch between the ideal free-standing equilibrium lattice of the crystal with respect to adjacent materials is included in the model by considering a discontinuous mismatch strain in the elastic energy. Since two-dimensional methods and the methods used in the previous literature where Dirichlet boundary conditions instead of the mismatch strain and only the wetting regime were considered, cannot be employed in this setting, we proceed differently, by including in the analysis the dewetting regime and carefully analyzing the fine properties of energy-equibounded sequences. This analysis allows to establish both a compactness property in the family of admissible configurations and the lower semicontinuity of the energy with respect to the topology induced by the L1-convergence of sets and a.e. convergence of displacement fields, so that the direct method can be applied. We also prove that our arguments work as well in the setting with Dirichlet boundary conditions.
{"title":"Existence of minimizers for the SDRI model in Rn: Wetting and dewetting regimes with mismatch strain","authors":"Shokhrukh Y. Kholmatov , Paolo Piovano","doi":"10.1016/j.na.2026.114061","DOIUrl":"10.1016/j.na.2026.114061","url":null,"abstract":"<div><div>The existence and the regularity results obtained in [41] for the variational model introduced in [40] to study the optimal shape of crystalline materials in the setting of stress-driven rearrangement instabilities (SDRI) are extended from two dimensions to any dimensions <em>n</em> ≥ 2. The energy is the sum of the elastic and the surface energy contributions, which cannot be decoupled, and depend on configurational pairs consisting of a set and a function that model the region occupied by the crystal and the bulk displacement field, respectively. By following the physical literature, the “driving stress” due to the mismatch between the ideal free-standing equilibrium lattice of the crystal with respect to adjacent materials is included in the model by considering a discontinuous mismatch strain in the elastic energy. Since two-dimensional methods and the methods used in the previous literature where Dirichlet boundary conditions instead of the mismatch strain and only the wetting regime were considered, cannot be employed in this setting, we proceed differently, by including in the analysis the dewetting regime and carefully analyzing the fine properties of energy-equibounded sequences. This analysis allows to establish both a compactness property in the family of admissible configurations and the lower semicontinuity of the energy with respect to the topology induced by the <em>L</em><sup>1</sup>-convergence of sets and a.e. convergence of displacement fields, so that the direct method can be applied. We also prove that our arguments work as well in the setting with Dirichlet boundary conditions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114061"},"PeriodicalIF":1.3,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078531","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 : 2026-01-21DOI: 10.1016/j.na.2026.114059
Claudianor O. Alves
In this work we use variational methods to prove the existence and concentration of nonnegative solutions for the following class of problemswhere Δ1 is the Laplacian operator, ϵ is a positive parameter, is a continuous function having a subcritical growth and is a continuous function with a local minimum.
{"title":"On existence of solutions to a class of problems involving the 1−Laplace operator in whole RN via penalization method","authors":"Claudianor O. Alves","doi":"10.1016/j.na.2026.114059","DOIUrl":"10.1016/j.na.2026.114059","url":null,"abstract":"<div><div>In this work we use variational methods to prove the existence and concentration of nonnegative solutions for the following class of problems<span><span><span><math><mrow><mo>−</mo><mi>ϵ</mi><msub><mstyle><mi>Δ</mi></mstyle><mn>1</mn></msub><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mfrac><mi>u</mi><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mfrac><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mi>R</mi><mi>N</mi></msup><mo>,</mo><mspace></mspace><mi>u</mi><mo>∈</mo><mi>B</mi><mi>V</mi><mrow><mo>(</mo><msup><mi>R</mi><mi>N</mi></msup><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where Δ<sub>1</sub> is the <span><math><mrow><mn>1</mn><mo>−</mo></mrow></math></span>Laplacian operator, ϵ is a positive parameter, <span><math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math></span> is a continuous function having a subcritical growth and <span><math><mrow><mi>V</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mi>N</mi></msup><mo>→</mo><mi>R</mi></mrow></math></span> is a continuous function with a local minimum.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114059"},"PeriodicalIF":1.3,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023412","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 : 2026-01-21DOI: 10.1016/j.na.2026.114058
Jordan Serres
We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-Émery criterion.
{"title":"Contractive transport maps from S2 to nearly spherical surfaces with positive Ricci curvature","authors":"Jordan Serres","doi":"10.1016/j.na.2026.114058","DOIUrl":"10.1016/j.na.2026.114058","url":null,"abstract":"<div><div>We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-Émery criterion.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114058"},"PeriodicalIF":1.3,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023414","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 : 2026-01-20DOI: 10.1016/j.na.2026.114056
Paulo Henryque C. Silva
In this paper, we establish two p-eigenvalue pinching sphere theorems, for the p-Laplacian, p > 1. The first result states that if the first non-zero p-eigenvalue of a closed Riemannian n-manifold with sectional curvature KM ≥ 1 is sufficiently close to the first non-zero p-eigenvalue of then M is homeomorphic to . The second states that if the first non-zero p-eigenvalue of a closed Riemannian n-manifold with Ricci curvature and injectivity radius injM ≥ i0 > 0 is sufficiently close to the first non-zero p-eigenvalue of then M is diffeomorphic to . Our results extend sphere theorems originally settled for the Laplacian by S. Croke [1] and G.P. Bessa [2] respectively.
{"title":"p-Eigenvalue pinching sphere theorems","authors":"Paulo Henryque C. Silva","doi":"10.1016/j.na.2026.114056","DOIUrl":"10.1016/j.na.2026.114056","url":null,"abstract":"<div><div>In this paper, we establish two <em>p</em>-eigenvalue pinching sphere theorems, for the <em>p</em>-Laplacian, <em>p</em> > 1. The first result states that if the first non-zero <em>p</em>-eigenvalue of a closed Riemannian <em>n</em>-manifold with sectional curvature <em>K<sub>M</sub></em> ≥ 1 is sufficiently close to the first non-zero <em>p</em>-eigenvalue of <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span> then <em>M</em> is homeomorphic to <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span>. The second states that if the first non-zero <em>p</em>-eigenvalue of a closed Riemannian <em>n</em>-manifold with Ricci curvature <span><math><mrow><msub><mrow><mrow><mi>R</mi></mrow><mi>i</mi><mi>c</mi></mrow><mi>M</mi></msub><mo>≥</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> and injectivity radius inj<sub><em>M</em></sub> ≥ <em>i</em><sub>0</sub> > 0 is sufficiently close to the first non-zero <em>p</em>-eigenvalue of <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span> then <em>M</em> is diffeomorphic to <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span>. Our results extend sphere theorems originally settled for the Laplacian by S. Croke [1] and G.P. Bessa [2] respectively.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114056"},"PeriodicalIF":1.3,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023413","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 : 2026-01-16DOI: 10.1016/j.na.2026.114055
Kyungkeun Kang , Jörg Wolf
We study the partial regularity of suitable weak solutions for non-Newtonian Navier-Stokes equations that is specifically a power-law type of shear thickening flows. We prove a generalization of CKN theorem for the power in the range . As one of our main tools, we establish an ϵ-regularity criterion, that is, the smallness of a scaling invariant local norm for of the velocity filed, which seems to be of independent interest.
{"title":"On the partial regularity of suitable weak solutions to the equations of shear thickening fluids","authors":"Kyungkeun Kang , Jörg Wolf","doi":"10.1016/j.na.2026.114055","DOIUrl":"10.1016/j.na.2026.114055","url":null,"abstract":"<div><div>We study the partial regularity of suitable weak solutions for non-Newtonian Navier-Stokes equations that is specifically a power-law type of shear thickening flows. We prove a generalization of CKN theorem for the power in the range <span><math><mrow><mo>[</mo><mn>2</mn><mo>,</mo><mfrac><mn>11</mn><mn>5</mn></mfrac><mo>)</mo></mrow></math></span>. As one of our main tools, we establish an ϵ-regularity criterion, that is, the smallness of a scaling invariant local norm for <span><math><mrow><msubsup><mi>L</mi><mi>t</mi><mi>∞</mi></msubsup><msubsup><mi>L</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math></span> of the velocity filed, which seems to be of independent interest.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114055"},"PeriodicalIF":1.3,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979448","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 : 2026-01-14DOI: 10.1016/j.na.2026.114057
Mostafa Meliani
We study the local existence of solutions to the Navier–Stokes–Fourier-magnetohydrodynamics (NSF-MHD) system describing the motion of a compressible, viscous, electrically and heat conducting fluid in the Lp–Lq class with inhomogeneous boundary conditions. The open system is allowed to receive incoming matter from the outside through (part of) the boundary which we refer to as an inflow boundary. This setup brings about a difficulty in estimating the regularity of the density ϱ which we remedy by assuming appropriate hypotheses on the velocity field, domain boundary and on the boundary and initial data of ϱ. The main result ensures the local well-posedness of the full NSF-MHD system which is shown through a linearization combined with a Banach fixed-point theorem.
{"title":"Lp–Lq existence for the open compressible MHD system","authors":"Mostafa Meliani","doi":"10.1016/j.na.2026.114057","DOIUrl":"10.1016/j.na.2026.114057","url":null,"abstract":"<div><div>We study the local existence of solutions to the Navier–Stokes–Fourier-magnetohydrodynamics (NSF-MHD) system describing the motion of a compressible, viscous, electrically and heat conducting fluid in the <em>L<sup>p</sup></em>–<em>L<sup>q</sup></em> class with inhomogeneous boundary conditions. The open system is allowed to receive incoming matter from the outside through (part of) the boundary which we refer to as an inflow boundary. This setup brings about a difficulty in estimating the regularity of the density ϱ which we remedy by assuming appropriate hypotheses on the velocity field, domain boundary and on the boundary and initial data of ϱ. The main result ensures the local well-posedness of the full NSF-MHD system which is shown through a linearization combined with a Banach fixed-point theorem.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114057"},"PeriodicalIF":1.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979449","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 : 2026-01-05DOI: 10.1016/j.na.2025.114054
Daniel Goodair
We obtain energy estimates for a transport and stretching noise under Leray Projection on a 2D bounded convex domain, in Sobolev Spaces of arbitrarily high order. The estimates are taken in equivalent inner products, defined through powers of the Stokes Operator with a specific choice of Navier boundary conditions. We exploit fine properties of the noise in relation to the Stokes Operator to achieve cancellation of derivatives in the presence of the Leray Projector. As a result, we achieve an additional degree of regularity in the corresponding Stochastic Navier-Stokes Equation to attain a true strong solution of the original Stratonovich equation. Furthermore for any order of smoothness, we can construct a strong solution of a hyperdissipative version of the Stochastic Navier-Stokes Equation with the given regularity; hyperdissipation is only required to control the nonlinear term in the presence of a boundary. We supplement the result by obtaining smoothness without hyperdissipation on the torus, in 2D and 3D on the lifetime of solutions.
{"title":"High order smoothness for stochastic Navier-Stokes equations with transport and stretching noise on bounded domains","authors":"Daniel Goodair","doi":"10.1016/j.na.2025.114054","DOIUrl":"10.1016/j.na.2025.114054","url":null,"abstract":"<div><div>We obtain energy estimates for a transport and stretching noise under Leray Projection on a 2D bounded convex domain, in Sobolev Spaces of arbitrarily high order. The estimates are taken in equivalent inner products, defined through powers of the Stokes Operator with a specific choice of Navier boundary conditions. We exploit fine properties of the noise in relation to the Stokes Operator to achieve cancellation of derivatives in the presence of the Leray Projector. As a result, we achieve an additional degree of regularity in the corresponding Stochastic Navier-Stokes Equation to attain a true strong solution of the original Stratonovich equation. Furthermore for any order of smoothness, we can construct a strong solution of a hyperdissipative version of the Stochastic Navier-Stokes Equation with the given regularity; hyperdissipation is only required to control the nonlinear term in the presence of a boundary. We supplement the result by obtaining smoothness without hyperdissipation on the torus, in 2D and 3D on the lifetime of solutions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114054"},"PeriodicalIF":1.3,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928323","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 : 2026-01-03DOI: 10.1016/j.na.2025.114046
Sergio Zamora , Xingyu Zhu
For a polycyclic group Λ, rank(Λ) is defined as the number of factors in a polycyclic decomposition of Λ. For a finitely generated group G, rank(G) is defined as the infimum of rank(Λ) among finite index polycyclic subgroups Λ ≤ G.
For a compact RCD(K, N) space with diam(X) ≤ ε(K, N), the rank of π1(X) is at most N. We show that in case of equality, X is homeomorphic to an infranilmanifold, generalizing a result by Kapovitch–Wilking to the non-smooth setting.
{"title":"Topological rigidity of small RCD(K,N) spaces with maximal rank","authors":"Sergio Zamora , Xingyu Zhu","doi":"10.1016/j.na.2025.114046","DOIUrl":"10.1016/j.na.2025.114046","url":null,"abstract":"<div><div>For a polycyclic group Λ, rank(Λ) is defined as the number of <span><math><mi>Z</mi></math></span> factors in a polycyclic decomposition of Λ. For a finitely generated group <em>G</em>, rank(<em>G</em>) is defined as the infimum of rank(Λ) among finite index polycyclic subgroups Λ ≤ <em>G</em>.</div><div>For a compact RCD(<em>K, N</em>) space <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></math></span> with diam(<em>X</em>) ≤ ε(<em>K, N</em>), the rank of <em>π</em><sub>1</sub>(<em>X</em>) is at most <em>N</em>. We show that in case of equality, <em>X</em> is homeomorphic to an infranilmanifold, generalizing a result by Kapovitch–Wilking to the non-smooth setting.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114046"},"PeriodicalIF":1.3,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886071","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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