arXiv - MATH - Analysis of PDEs最新文献_第2页

Contact discontinuities for 2-D isentropic Euler are unique in 1-D but wildly non-unique otherwise 二维等熵欧拉接触间断点在一维中是唯一的，但在其他维度中则是非唯一的

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-17 DOI: arxiv-2409.11296

Sam G. Krupa, László Székelyhidi Jr

{"title":"Contact discontinuities for 2-D isentropic Euler are unique in 1-D but wildly non-unique otherwise","authors":"Sam G. Krupa, László Székelyhidi Jr","doi":"arxiv-2409.11296","DOIUrl":"https://doi.org/arxiv-2409.11296","url":null,"abstract":"We develop a general framework for studying non-uniqueness of the Riemann\u0000problem for the isentropic compressible Euler system in two spatial dimensions,\u0000and in this paper we present the most delicate result of our method:\u0000non-uniqueness of the contact discontinuity. Our approach is computational, and\u0000uses the pressure law as an additional degree of freedom. The stability of the contact discontinuities for this system is a major open\u0000problem (see Gui-Qiang Chen and Ya-Guang Wang [Nonlinear partial differential\u0000equations, volume 7 of Abel Symposia. Springer, Heidelberg, 2012.]). We find a smooth pressure law $p$, verifying the physically relevant\u0000condition $p'>0$, such that for the isentropic compressible Euler system with\u0000this pressure law, contact discontinuity initial data is wildly non-unique in\u0000the class of bounded, admissible weak solutions. This result resolves the\u0000question of uniqueness for contact discontinuity solutions in the compressible\u0000regime. Moreover, in the same regularity class in which we have non-uniqueness of the\u0000contact discontinuity, i.e. $L^infty$, with no $BV$ regularity or\u0000self-similarity, we show that the classical contact discontinuity solution to\u0000the two-dimensional isentropic compressible Euler system is in fact unique in\u0000the class of bounded, admissible weak solutions if we restrict to 1-D\u0000solutions.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A limiting case of a theorem of C. Miranda for layer potentials in Schauder spaces C. Miranda 关于 Schauder 空间中层势的定理的极限情况

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-17 DOI: arxiv-2409.11132

Massimo Lanza de Cristoforis

引用次数: 0

Global solutions to 3D quadratic nonlinear Schrödinger-type equation 三维二次非线性薛定谔型方程的全局解

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-17 DOI: arxiv-2409.10804

Zihua Guo, Naijia Liu, Liang Song

引用次数: 0

A note on the radially symmetry in the moving plane method 关于移动平面法径向对称性的说明

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-17 DOI: arxiv-2409.10834

Shu-Yu Hsu

{"title":"A note on the radially symmetry in the moving plane method","authors":"Shu-Yu Hsu","doi":"arxiv-2409.10834","DOIUrl":"https://doi.org/arxiv-2409.10834","url":null,"abstract":"Let $Omegasubsetmathbb{R}^n$, $nge 2$, be a bounded connected $C^2$\u0000domain. For any unit vector $nuinmathbb{R}^n$, let\u0000$T_{lambda}^{nu}={xinmathbb{R}^n:xcdotnu=lambda}$,\u0000$Sigma_{lambda}^{nu}={xinOmega:xcdotnu<lambda}$ and\u0000$x^{ast}=x-2(xcdotnu-lambda)nu$ be the reflection of a point\u0000$xinmathbb{R}^n$ about the plane $T_{lambda}^{nu}$. Let\u0000$widetilde{Sigma}_{lambda}^{nu}={xinOmega:x^{ast}inSigma_{lambda}^{nu}}$\u0000and $uin C^2(overline{Omega})$. Suppose for any unit vector\u0000$nuinmathbb{R}^n$, there exists a constant $lambda_{nu}inmathbb{R}$ such\u0000that $Omega$ is symmetric about the plane $T_{lambda_{nu}}^{nu}$ and $u$ is\u0000symmetric about the plane $T_{lambda_{nu}}^{nu}$ and satisfies\u0000(i)$,frac{partial u}{partialnu}(x)>0quadforall xin\u0000Sigma_{lambda_{nu}}^{nu}$ and (ii)$,frac{partial\u0000u}{partialnu}(x)<0quadforall xin\u0000widetilde{Sigma}_{lambda_{nu}}^{nu}$. We will give a simple proof that $u$\u0000is radially symmetric about some point $x_0inOmega$ and $Omega$ is a ball\u0000with center at $x_0$. Similar result holds for the domain $mathbb{R}^n$ and\u0000function $uin C^2(mathbb{R}^n)$ satisfying similar monotonicity and symmetry\u0000conditions. We also extend this result under weaker hypothesis on the function\u0000$u$.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Existence of an extremal function of Sobolev critical embedding with an $α$-homogeneous weight 具有α$同调权重的索波列夫临界嵌入极值函数的存在性

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-17 DOI: arxiv-2409.11193

Petr Gurka, Daniel Hauer

引用次数: 0

A Physics Informed Neural Network (PINN) Methodology for Coupled Moving Boundary PDEs 耦合移动边界 PDE 的物理信息神经网络 (PINN) 方法学

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-17 DOI: arxiv-2409.10910

Shivprasad KathaneIndian Institute of Technology Bombay Mumbai India, Shyamprasad KaragaddeIndian Institute of Technology Bombay Mumbai India

{"title":"A Physics Informed Neural Network (PINN) Methodology for Coupled Moving Boundary PDEs","authors":"Shivprasad KathaneIndian Institute of Technology Bombay Mumbai India, Shyamprasad KaragaddeIndian Institute of Technology Bombay Mumbai India","doi":"arxiv-2409.10910","DOIUrl":"https://doi.org/arxiv-2409.10910","url":null,"abstract":"Physics-Informed Neural Network (PINN) is a novel multi-task learning\u0000framework useful for solving physical problems modeled using differential\u0000equations (DEs) by integrating the knowledge of physics and known constraints\u0000into the components of deep learning. A large class of physical problems in\u0000materials science and mechanics involve moving boundaries, where interface flux\u0000balance conditions are to be satisfied while solving DEs. Examples of such\u0000systems include free surface flows, shock propagation, solidification of pure\u0000and alloy systems etc. While recent research works have explored applicability\u0000of PINNs for an uncoupled system (such as solidification of pure system), the\u0000present work reports a PINN-based approach to solve coupled systems involving\u0000multiple governing parameters (energy and species, along with multiple\u0000interface balance equations). This methodology employs an architecture\u0000consisting of a separate network for each variable with a separate treatment of\u0000each phase, a training strategy which alternates between temporal learning and\u0000adaptive loss weighting, and a scheme which progressively reduces the\u0000optimisation space. While solving the benchmark problem of binary alloy\u0000solidification, it is distinctly successful at capturing the complex\u0000composition profile, which has a characteristic discontinuity at the interface\u0000and the resulting predictions align well with the analytical solutions. The\u0000procedure can be generalised for solving other transient multiphysics problems\u0000especially in the low-data regime and in cases where measurements can reveal\u0000new physics.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Existence of multiple radial solutions for nonlinear equation involving the mean curvature operator in Lorentz-Minkowski space 洛伦兹-闵科夫斯基空间中涉及平均曲率算子的非线性方程存在多个径向解

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-17 DOI: arxiv-2409.11039

Vittorio Coti Zelati, Xu Dong, Yuanhong Wei

引用次数: 0

Output-feedback stabilization of an underactuated network of N interconnected n + m hyperbolic PDE systems 由 N 个相互连接的 n + m 个双曲 PDE 系统组成的欠激励网络的输出反馈稳定问题

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10087

Jean AuriolL2S

{"title":"Output-feedback stabilization of an underactuated network of N interconnected n + m hyperbolic PDE systems","authors":"Jean AuriolL2S","doi":"arxiv-2409.10087","DOIUrl":"https://doi.org/arxiv-2409.10087","url":null,"abstract":"In this article, we detail the design of an output feedback stabilizing\u0000control law for an underactuated network of N subsystems of n + m\u0000heterodirectional linear first-order hyperbolic Partial Differential Equations\u0000interconnected through their boundaries. The network has a chain structure, as\u0000only one of the subsystems is actuated. The available measurements are located\u0000at the opposite extremity of the chain. The proposed approach introduces a new\u0000type of integral transformation to tackle in-domain couplings in the different\u0000subsystems while guaranteeing a ''clear actuation path'' between the control\u0000input and the different subsystems. Then, it is possible to state several\u0000essential properties of each subsystem: output trajectory tracking,\u0000input-to-state stability, and predictability (the possibility of designing a\u0000state prediction). We recursively design a stabilizing state-feedback\u0000controller by combining these properties. We then design a state-observer that\u0000reconstructs delayed values of the states. This observer is combined with the\u0000state-feedback control law to obtain an output-feedback controller. Simulations\u0000complete the presentation.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Scattering for the generalized Hartree equation with a potential 广义哈特里方程的势散射

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10769

Carlos M. Guzmán, Cristian Loli, Luis P. Yapu

引用次数: 0

Continuity of the linearized forward map of electrical impedance tomography from square-integrable perturbations to Hilbert-Schmidt operators 从方整扰动到希尔伯特-施密特算子的电阻抗断层扫描线性化正向图的连续性

arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10671

Joanna Bisch, Markus Hirvensalo, Nuutti Hyvönen

引用次数: 0