Pub Date : 2024-10-01DOI: 10.1016/j.na.2024.113668
Yanqing Wang , Jing Yang , Yulin Ye
In this paper, we are concerned with the conservation of energy and magnetic helicity of weak solutions for both the electron and Hall magnetohydrodynamic equations. Various energy and magnetic helicity conservation criteria in Onsager’s critical spaces and in these systems are established. Furthermore, we observe that the conservation criteria for energy and magnetic helicity in the EMHD equations correspond to the helicity and energy conservation principles in ideal incompressible Euler equations, respectively.
{"title":"On two conserved quantities in the inviscid electron and Hall magnetohydrodynamic equations","authors":"Yanqing Wang , Jing Yang , Yulin Ye","doi":"10.1016/j.na.2024.113668","DOIUrl":"10.1016/j.na.2024.113668","url":null,"abstract":"<div><div>In this paper, we are concerned with the conservation of energy and magnetic helicity of weak solutions for both the electron and Hall magnetohydrodynamic equations. Various energy and magnetic helicity conservation criteria in Onsager’s critical spaces <span><math><msubsup><mrow><munder><mrow><mi>B</mi></mrow><mo>̲</mo></munder></mrow><mrow><mi>p</mi><mo>,</mo><mi>V</mi><mi>M</mi><mi>O</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>c</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msubsup></math></span> in these systems are established. Furthermore, we observe that the conservation criteria for energy and magnetic helicity in the EMHD equations correspond to the helicity and energy conservation principles in ideal incompressible Euler equations, respectively.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113668"},"PeriodicalIF":1.3,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142359485","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 : 2024-09-30DOI: 10.1016/j.na.2024.113676
Miroslav Bulíček , Jakub Woźnicki
We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or data. The key difficulty consists of the presence of a monotone operator subjected to a non-standard growth condition, controlled by the exponent depending on the time and the spatial variable. We show the existence of a weak and an entropy solution to our system, as well as the uniqueness of an entropy solution, under the assumption of boundedness and log-Hölder continuity of the variable exponent with respect to the spatial variable. On the other hand, we do not assume any smoothness of with respect to the time variable.
我们考虑的是一个具有 Dirichlet 边界条件和测量或 L1 数据的抛物线偏微分方程。主要难点在于存在一个单调算子 A，该算子受制于非标准增长条件，由取决于时间和空间变量的指数 p 控制。我们证明了我们系统的弱解和熵解的存在性，以及熵解的唯一性，前提是变量指数 p 相对于空间变量具有有界性和 log-Hölder 连续性。另一方面，我们不假设 p 相对于时间变量的平滑性。
{"title":"Parabolic equations with non-standard growth and measure or integrable data","authors":"Miroslav Bulíček , Jakub Woźnicki","doi":"10.1016/j.na.2024.113676","DOIUrl":"10.1016/j.na.2024.113676","url":null,"abstract":"<div><div>We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> data. The key difficulty consists of the presence of a monotone operator <span><math><mi>A</mi></math></span> subjected to a non-standard growth condition, controlled by the exponent <span><math><mi>p</mi></math></span> depending on the time and the spatial variable. We show the existence of a weak and an entropy solution to our system, as well as the uniqueness of an entropy solution, under the assumption of boundedness and log-Hölder continuity of the variable exponent <span><math><mi>p</mi></math></span> with respect to the spatial variable. On the other hand, we do not assume any smoothness of <span><math><mi>p</mi></math></span> with respect to the time variable.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113676"},"PeriodicalIF":1.3,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-30DOI: 10.1016/j.na.2024.113656
M. Fortuna, A. Garroni
We prove an homogenization result, in terms of -convergence, for energies concentrated on rectifiable lines in without boundary. The main application of our result is in the context of dislocation lines in dimension 3. The result presented here shows that the line tension energy of unions of single line defects converge to the energy associated to macroscopic densities of dislocations carrying plastic deformation. As a byproduct of our construction for the upper bound for the -Limit, we obtain an alternative proof of the density of rectifiable 1-currents without boundary in the space of divergence free fields.
{"title":"Homogenization of line tension energies","authors":"M. Fortuna, A. Garroni","doi":"10.1016/j.na.2024.113656","DOIUrl":"10.1016/j.na.2024.113656","url":null,"abstract":"<div><div>We prove an homogenization result, in terms of <span><math><mi>Γ</mi></math></span>-convergence, for energies concentrated on rectifiable lines in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> without boundary. The main application of our result is in the context of dislocation lines in dimension 3. The result presented here shows that the line tension energy of unions of single line defects converge to the energy associated to macroscopic densities of dislocations carrying plastic deformation. As a byproduct of our construction for the upper bound for the <span><math><mi>Γ</mi></math></span>-Limit, we obtain an alternative proof of the density of rectifiable 1-currents without boundary in the space of divergence free fields.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113656"},"PeriodicalIF":1.3,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142359484","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 : 2024-09-29DOI: 10.1016/j.na.2024.113667
José M. Arrieta , Jean Carlos Nakasato , Manuel Villanueva-Pesqueira
This paper presents an extension of the unfolding operator technique, initially applied to two-dimensional domains, to the realm of three-dimensional thin domains. The advancement of this methodology is pivotal, as it enhances our understanding and analysis of three-dimensional geometries, which are crucial in various practical fields such as engineering and physics. Our work delves into the asymptotic behavior of solutions to a reaction–diffusion equation with Neumann boundary conditions set within such a oscillatory 3-dimensional thin domain. The method introduced enables the deduction of effective problems across all scenarios, tackling the intrinsic complexity of these domains. This complexity is especially pronounced due to the possibility of diverse types of oscillations occurring along their boundaries.
{"title":"Homogenization in 3D thin domains with oscillating boundaries of different orders","authors":"José M. Arrieta , Jean Carlos Nakasato , Manuel Villanueva-Pesqueira","doi":"10.1016/j.na.2024.113667","DOIUrl":"10.1016/j.na.2024.113667","url":null,"abstract":"<div><div>This paper presents an extension of the unfolding operator technique, initially applied to two-dimensional domains, to the realm of three-dimensional thin domains. The advancement of this methodology is pivotal, as it enhances our understanding and analysis of three-dimensional geometries, which are crucial in various practical fields such as engineering and physics. Our work delves into the asymptotic behavior of solutions to a reaction–diffusion equation with Neumann boundary conditions set within such a oscillatory 3-dimensional thin domain. The method introduced enables the deduction of effective problems across all scenarios, tackling the intrinsic complexity of these domains. This complexity is especially pronounced due to the possibility of diverse types of oscillations occurring along their boundaries.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113667"},"PeriodicalIF":1.3,"publicationDate":"2024-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425075","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 : 2024-09-26DOI: 10.1016/j.na.2024.113672
Emil Dare , Markus Kiderlen , Christoph Thäle
Consider a uniformly distributed random linear subspace and a stochastically independent random affine subspace in , both of fixed dimension. For a natural class of distributions for we show that the intersection admits a density with respect to the invariant measure. This density depends only on the distance of to the origin and is derived explicitly. It can be written as the product of a power of and a part involving an incomplete beta integral. Choosing uniformly among all affine subspaces of fixed dimension hitting the unit ball, we derive an explicit density for the random variable and study the behavior of the probability that hits the unit ball in high dimensions. Lastly, we show that our result can be extended to the setting where is tangent to the unit sphere, in which case we again derive the density for . Our probabilistic results are derived by means of a new integral–geometric transformation formula of Blaschke–Petkantschin type.
考虑 Rn 中的均匀分布随机线性子空间 L 和随机独立随机仿射子空间 E，两者的维数都是固定的。对于 E 的一类自然分布，我们证明 L∩E 的交集有一个关于不变度量的密度。这个密度只取决于 L∩E 到原点的距离 d(o,E∩L)，并且是明确推导出来的。它可以写成 d(o,E∩L)的幂与不完全贝塔积分的乘积。我们在所有固定维度的仿射子空间中均匀地选择 E，得出了随机变量 d(o,E∩L)的显式密度，并研究了 E∩L 在高维度上击中单位球的概率行为。最后，我们证明我们的结果可以扩展到 E 与单位球相切的情况，在这种情况下，我们再次推导出 d(o,E∩L) 的密度。我们的概率结果是通过布拉什克-佩特康钦类型的新积分几何变换公式得出的。
{"title":"A Blaschke–Petkantschin formula for linear and affine subspaces with application to intersection probabilities","authors":"Emil Dare , Markus Kiderlen , Christoph Thäle","doi":"10.1016/j.na.2024.113672","DOIUrl":"10.1016/j.na.2024.113672","url":null,"abstract":"<div><div>Consider a uniformly distributed random linear subspace <span><math><mi>L</mi></math></span> and a stochastically independent random affine subspace <span><math><mi>E</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, both of fixed dimension. For a natural class of distributions for <span><math><mi>E</mi></math></span> we show that the intersection <span><math><mrow><mi>L</mi><mo>∩</mo><mi>E</mi></mrow></math></span> admits a density with respect to the invariant measure. This density depends only on the distance <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><mi>L</mi><mo>∩</mo><mi>E</mi></mrow></math></span> to the origin and is derived explicitly. It can be written as the product of a power of <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span> and a part involving an incomplete beta integral. Choosing <span><math><mi>E</mi></math></span> uniformly among all affine subspaces of fixed dimension hitting the unit ball, we derive an explicit density for the random variable <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span> and study the behavior of the probability that <span><math><mrow><mi>E</mi><mo>∩</mo><mi>L</mi></mrow></math></span> hits the unit ball in high dimensions. Lastly, we show that our result can be extended to the setting where <span><math><mi>E</mi></math></span> is tangent to the unit sphere, in which case we again derive the density for <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>o</mi><mo>,</mo><mi>E</mi><mo>∩</mo><mi>L</mi><mo>)</mo></mrow></mrow></math></span>. Our probabilistic results are derived by means of a new integral–geometric transformation formula of Blaschke–Petkantschin type.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113672"},"PeriodicalIF":1.3,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-26DOI: 10.1016/j.na.2024.113674
Sergey Shindin, Nabendra Parumasur
We show that the Benjamin equation is globally well-posed for real-valued data in the weighted space where and . The proof is based on direct extensions of standard linear and bilinear estimates originated in Kenig et al. (1993), Kenig et al. (1996), Linares (1999), Kozono et al. (2001), Colliander et al. (2003), Li and Wu (2010) to the weighted settings.
{"title":"Global low regularity solutions to the Benjamin equation in weighted spaces","authors":"Sergey Shindin, Nabendra Parumasur","doi":"10.1016/j.na.2024.113674","DOIUrl":"10.1016/j.na.2024.113674","url":null,"abstract":"<div><div>We show that the Benjamin equation is globally well-posed for real-valued data in the weighted space <span><span><span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>∩</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>r</mi></mrow><mrow><mi>s</mi><mo>−</mo><mn>2</mn><mi>r</mi></mrow></msubsup><mo>≔</mo><mrow><mo>{</mo><mrow><mi>u</mi><mspace></mspace><mo>|</mo><mspace></mspace><msub><mrow><mo>‖</mo><mi>u</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><mover><mrow><mi>u</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>ξ</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>,</mo><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mi>s</mi><mo>−</mo><mn>2</mn><mi>r</mi><mo>)</mo></mrow></mrow></msup><mi>d</mi><mi>ξ</mi><mo>)</mo></mrow></mrow></msub><mo><</mo><mi>∞</mi></mrow><mo>}</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mn>0</mn><mo>≤</mo><mi>r</mi></mrow></math></span> and <span><math><mrow><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mi>r</mi><mo><</mo><mi>s</mi></mrow></math></span>. The proof is based on direct extensions of standard linear and bilinear estimates originated in Kenig et al. (1993), Kenig et al. (1996), Linares (1999), Kozono et al. (2001), Colliander et al. (2003), Li and Wu (2010) to the weighted settings.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113674"},"PeriodicalIF":1.3,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-23DOI: 10.1016/j.na.2024.113670
Hongxia Xue, Jianwei Dong
In this paper, we study the free boundary problem of an inviscid two-phase model, where we take the pressure function as (, ) with and being the densities of two phases. First, we construct some self-similar analytical solutions for the -dimensional radially symmetric case by using some ansatzs, and investigate the spreading rate of the free boundary by using the method of averaged quantities. Second, we extend the results of the -dimensional radially symmetric case to the three-dimensional cylindrically symmetric case. Third, we present some analytical solutions for the three-dimensional cylindrically symmetric model with a Coriolis force. From the analytical solutions constructed in this paper, we find that the Coriolis force can prevent the free boundary from spreading out infinitely.
本文研究了不粘性两相模型的自由边界问题，其中压力函数为 P(n,ρ)=ργ+nα (γ>1, α≥1)，n 和 ρ 分别为两相的密度。首先，我们利用一些解析式构建了 N 维径向对称情况下的一些自相似解析解，并利用平均量方法研究了自由边界的扩散率。其次，我们将 N 维径向对称情况的结果扩展到三维圆柱对称情况。第三，我们给出了具有科里奥利力的三维圆柱对称模型的一些解析解。从本文构建的解析解中，我们发现科里奥利力可以阻止自由边界无限向外扩展。
{"title":"Analytical solutions to the free boundary problem of a two-phase model with radial and cylindrical symmetry","authors":"Hongxia Xue, Jianwei Dong","doi":"10.1016/j.na.2024.113670","DOIUrl":"10.1016/j.na.2024.113670","url":null,"abstract":"<div><div>In this paper, we study the free boundary problem of an inviscid two-phase model, where we take the pressure function as <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>ρ</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mi>γ</mi></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>γ</mi><mo>></mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>α</mi><mo>≥</mo><mn>1</mn></mrow></math></span>) with <span><math><mi>n</mi></math></span> and <span><math><mi>ρ</mi></math></span> being the densities of two phases. First, we construct some self-similar analytical solutions for the <span><math><mi>N</mi></math></span>-dimensional radially symmetric case by using some ansatzs, and investigate the spreading rate of the free boundary by using the method of averaged quantities. Second, we extend the results of the <span><math><mi>N</mi></math></span>-dimensional radially symmetric case to the three-dimensional cylindrically symmetric case. Third, we present some analytical solutions for the three-dimensional cylindrically symmetric model with a Coriolis force. From the analytical solutions constructed in this paper, we find that the Coriolis force can prevent the free boundary from spreading out infinitely.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113670"},"PeriodicalIF":1.3,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001895/pdfft?md5=f5c63e293b0091e1cef7e731ee5a5250&pid=1-s2.0-S0362546X24001895-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider stochastic thin-film equation with nonlinear drift terms, colored Gaussian Stratonovich noise, as well as nonlinear colored Wiener noise. By means of Trotter–Kato-type decomposition into deterministic and stochastic parts, we couple both of these dynamics via a discrete-in-time scheme, and establish its convergence to a non-negative weak martingale solution.
{"title":"Thin film equations with nonlinear deterministic and stochastic perturbations","authors":"Oleksiy Kapustyan , Olha Martynyuk , Oleksandr Misiats , Oleksandr Stanzhytskyi","doi":"10.1016/j.na.2024.113646","DOIUrl":"10.1016/j.na.2024.113646","url":null,"abstract":"<div><p>In this paper we consider stochastic thin-film equation with nonlinear drift terms, colored Gaussian Stratonovich noise, as well as nonlinear colored Wiener noise. By means of Trotter–Kato-type decomposition into deterministic and stochastic parts, we couple both of these dynamics via a discrete-in-time scheme, and establish its convergence to a non-negative weak martingale solution.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113646"},"PeriodicalIF":1.3,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001652/pdfft?md5=f3d21fbe23f0caa335b0a9f697a81c70&pid=1-s2.0-S0362546X24001652-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-16DOI: 10.1016/j.na.2024.113669
Jingwen Ji , Haiyun Deng , Feida Jiang
In this paper, we establish a necessary and sufficient condition for the solvability of the real Monge–Ampère equation in bounded domains with infinite Dirichlet boundary condition. The Monge–Ampère operator is derived from geometry and has recently received much attention. Our result embraces the case where is positive and satisfies the Keller–Osserman type condition. We describe the asymptotic behavior of the solution by constructing suitable sub-solutions and super-solutions, and obtain a uniqueness result in star-shaped domains by using a scaling technique.
{"title":"On the boundary blow-up problem for real (n−1) Monge–Ampère equation","authors":"Jingwen Ji , Haiyun Deng , Feida Jiang","doi":"10.1016/j.na.2024.113669","DOIUrl":"10.1016/j.na.2024.113669","url":null,"abstract":"<div><p>In this paper, we establish a necessary and sufficient condition for the solvability of the real <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> Monge–Ampère equation <span><math><mrow><mover><mrow><mo>det</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></mover><mrow><mo>(</mo><mi>Δ</mi><mi>u</mi><mi>I</mi><mo>−</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> in bounded domains with infinite Dirichlet boundary condition. The <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span> Monge–Ampère operator is derived from geometry and has recently received much attention. Our result embraces the case <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>h</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span> is positive and <span><math><mi>f</mi></math></span> satisfies the Keller–Osserman type condition. We describe the asymptotic behavior of the solution by constructing suitable sub-solutions and super-solutions, and obtain a uniqueness result in star-shaped domains by using a scaling technique.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113669"},"PeriodicalIF":1.3,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001883/pdfft?md5=dcb2b703c48c88a6c661fc63e5774351&pid=1-s2.0-S0362546X24001883-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-13DOI: 10.1016/j.na.2024.113665
Qiang Du , James M. Scott , Zirui Xu
We study the minimizers of a degenerate case of the Ohta–Kawasaki energy, defined as the sum of the perimeter and a Coulombic nonlocal term. We start by investigating radially symmetric candidates which give us insights into the asymptotic behaviors of energy minimizers in the large mass limit. In order to numerically study the problems that are analytically challenging, we propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model. Our phase-field simulations and asymptotic results suggest that the energy minimizers exhibit behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes.
{"title":"Ohta–Kawasaki energy for amphiphiles: Asymptotics and phase-field simulations","authors":"Qiang Du , James M. Scott , Zirui Xu","doi":"10.1016/j.na.2024.113665","DOIUrl":"10.1016/j.na.2024.113665","url":null,"abstract":"<div><p>We study the minimizers of a degenerate case of the Ohta–Kawasaki energy, defined as the sum of the perimeter and a Coulombic nonlocal term. We start by investigating radially symmetric candidates which give us insights into the asymptotic behaviors of energy minimizers in the large mass limit. In order to numerically study the problems that are analytically challenging, we propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model. Our phase-field simulations and asymptotic results suggest that the energy minimizers exhibit behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"250 ","pages":"Article 113665"},"PeriodicalIF":1.3,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001846/pdfft?md5=c1e8f89875e4f78c3319ad9c2f245a48&pid=1-s2.0-S0362546X24001846-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142171934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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