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Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity 具有空间增长非线性的半线性波方程的好求和线性化
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-13 DOI: arxiv-2409.08594
Dhouha Draouil, Mohamed Majdoub
{"title":"Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity","authors":"Dhouha Draouil, Mohamed Majdoub","doi":"arxiv-2409.08594","DOIUrl":"https://doi.org/arxiv-2409.08594","url":null,"abstract":"We investigate the initial value problem for a defocusing semi-linear wave\u0000equation with spatially growing nonlinearity. Our analysis leads to global\u0000well-posedness in the energy space. Furthermore, we obtain the linearization of\u0000energy-bounded solutions using the methodology outlined in cite{P.GR}. The\u0000proof hinges on Moser-Trudinger type inequalities and Strichartz estimates.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260718","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
Regularity results for minimizers of non-autonomous integral functionals 非自治积分函数最小值的正则性结果
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-13 DOI: arxiv-2409.08796
Antonio Giuseppe Grimaldi, Stefania Russo
{"title":"Regularity results for minimizers of non-autonomous integral functionals","authors":"Antonio Giuseppe Grimaldi, Stefania Russo","doi":"arxiv-2409.08796","DOIUrl":"https://doi.org/arxiv-2409.08796","url":null,"abstract":"We establish the higher fractional differentiability for the minimizers of\u0000non-autonomous integral functionals of the form begin{equation} mathcal{F}(u,Omega):=int_Omega left[ f(x,Du)- g cdot u right] dx ,\u0000notag end{equation} under $(p,q)$-growth conditions. Besides a suitable\u0000differentiability assumption on the partial map $x mapsto D_xi f(x,xi)$, we\u0000do not need to assume any differentiability assumption on the function $g$.\u0000Moreover, we show that the higher differentiability result holds true also\u0000assuming strict convexity and growth conditions on $f$ only at infinity.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"207 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260710","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
Phaseless uniqueness for determining internal source in photo-thermal effect 确定光热效应内源的无相唯一性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-13 DOI: arxiv-2409.08532
Li-Ping Deng, Hongyu Liu, Zhi-Qiang Miao, Guang-Hui Zheng
{"title":"Phaseless uniqueness for determining internal source in photo-thermal effect","authors":"Li-Ping Deng, Hongyu Liu, Zhi-Qiang Miao, Guang-Hui Zheng","doi":"arxiv-2409.08532","DOIUrl":"https://doi.org/arxiv-2409.08532","url":null,"abstract":"The paper investigates an inverse problem of recovering the internal source\u0000from external temperature measurements in photo-thermal effect. The\u0000photo-thermal effect actually involves two physical processes: electromagnetic\u0000scattering and heat transfer, described by a nonlinear coupled system of\u0000Maxwell's equation and the heat transfer equation. The nonlinear coupling term\u0000in the system is represented by the square of the modulus of the\u0000electromagnetic (missing the phase information of the electromagnetic field),\u0000and the absence of this phase information poses a significant challenge to the\u0000reconstruction of the internal source. In addition, the interaction and mutual\u0000influence of multiple physical fields, including electric field, magnetic field\u0000and temperature field, add to the complexity involved in the inversion of the\u0000internal source. Based on the potential theory and asymptotic analysis, we\u0000prove that the internal source can be uniquely determined up to sign by the\u0000external temperature field. This provides a solid theoretical basis for\u0000designing the internal source inversion algorithm and further exploring the\u0000theoretical aspects of photo-thermal effect.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260719","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
Modified Wasserstein gradient flow formulation of time-fractional porous medium equations with nonlocal pressure 具有非局部压力的时间分数多孔介质方程的修正瓦瑟斯坦梯度流计算公式
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-13 DOI: arxiv-2409.08441
Nhan-Phu Chung, Thanh-Son Trinh
{"title":"Modified Wasserstein gradient flow formulation of time-fractional porous medium equations with nonlocal pressure","authors":"Nhan-Phu Chung, Thanh-Son Trinh","doi":"arxiv-2409.08441","DOIUrl":"https://doi.org/arxiv-2409.08441","url":null,"abstract":"We consider a class of time-fractional porous medium equations with nonlocal\u0000pressure. We show the existence of their weak solutions by proposing a JKO\u0000scheme for modified Wasserstein distance and a square fractional Sobolev norm.\u0000Moreover, the regularization effect and the Lp norm estimate are established in\u0000this paper.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260722","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
On a class of coupled obstacle systems 关于一类耦合障碍系统
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-13 DOI: arxiv-2409.08478
Lili Du, Xu Tang, Cong Wang
{"title":"On a class of coupled obstacle systems","authors":"Lili Du, Xu Tang, Cong Wang","doi":"arxiv-2409.08478","DOIUrl":"https://doi.org/arxiv-2409.08478","url":null,"abstract":"In this paper, we explore cooperative and competitive coupled obstacle\u0000systems, which, up to now, are new type obstacle systems and formed by coupling\u0000two equations belonging to classical obstacle problem. On one hand, applying\u0000the constrained minimizer in variational methods we establish the existence of\u0000solutions for the systems. Moreover, the optimal regularity of solutions is\u0000obtained, which is the cornerstone for further research on so-called free\u0000boundary. Furthermore, as coefficient $lambdato0$, there exists a sequence of\u0000solutions converging to solutions of the single classical obstacle equation. On\u0000the other hand, motivated by the heartstirring ideas of single classical\u0000obstacle problem, based on the corresponding blowup methods, Weiss type\u0000monotonicity formula and Monneau type monotonicity formula of systems to be\u0000studied, we investigate the regularity of free boundary, and on the regular and\u0000singular points in particular, as it should be, which is more challenging but\u0000exceedingly meaningful in solving free boundary problems.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260720","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
$mathscr{A}$-free truncation and higher integrability of minimisers 无$mathscr{A}$截断和最小值的高积分性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-13 DOI: arxiv-2409.08713
Stefan Schiffer
{"title":"$mathscr{A}$-free truncation and higher integrability of minimisers","authors":"Stefan Schiffer","doi":"arxiv-2409.08713","DOIUrl":"https://doi.org/arxiv-2409.08713","url":null,"abstract":"We show higher integrability of minimisers of functionals [ I(u) = int_{Omega} f(x,u(x)) ~mathrm{d}x ] subject to a differential constraint $mathscr{A} u=0$ under natural\u0000$p$-growth and $p$-coercivity conditions for $f$ and regularity assumptions on\u0000$Omega$. For the differential operator $mathscr{A}$ we asssume a rather\u0000abstract truncation property that, for instance, holds for operators\u0000$mathscr{A}=mathrm{curl}$ and $mathscr{A}=mathrm{div}$. The proofs are\u0000based on the comparison of the minimiser to the truncated version of the\u0000minimiser.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260716","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
Homogenisation of vectorial free-discontinuity functionals with cohesive type surface terms 带有内聚型表面项的矢量自由不连续函数的均质化
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-12 DOI: arxiv-2409.07820
Gianni Dal Maso, Davide Donati
{"title":"Homogenisation of vectorial free-discontinuity functionals with cohesive type surface terms","authors":"Gianni Dal Maso, Davide Donati","doi":"arxiv-2409.07820","DOIUrl":"https://doi.org/arxiv-2409.07820","url":null,"abstract":"The results on $Gamma$-limits of sequences of free-discontinuity functionals\u0000with bounded cohesive surface terms are extended to the case of vector-valued\u0000functions. In this framework, we prove an integral representation result for\u0000the $Gamma$-limit, which is then used to study deterministic and stochastic\u0000homogenisation problems for this type of functionals.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"160 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179014","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
Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance 低于施特劳斯指数的非线性薛定谔方程在加权空间中的全局好求和散射，无规规不变性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-12 DOI: arxiv-2409.08432
Masaki Kawamoto, Satoshi Masaki, Hayato Miyazaki
{"title":"Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance","authors":"Masaki Kawamoto, Satoshi Masaki, Hayato Miyazaki","doi":"arxiv-2409.08432","DOIUrl":"https://doi.org/arxiv-2409.08432","url":null,"abstract":"In this paper, we consider the nonlinear Schr\"{o}dinger equation with a\u0000general homogeneous nonlinearity in dimensions up to three. We assume that the\u0000degree (i.e., power) of the nonlinearity is such that the equation is\u0000mass-subcritical and short-range. We establish global well-posedness (GWP) and\u0000scattering for small data in the standard weighted space for a class of\u0000homogeneous nonlinearities, including non-gauge-invariant ones. Additionally,\u0000we include the case where the degree is less than or equal to the Strauss\u0000exponent. When the nonlinearity is not gauge-invariant, the standard Duhamel\u0000formulation fails to work effectively in the weighted Sobolev space; for\u0000instance, the Duhamel term may not be well-defined as a Bochner integral. To\u0000address this issue, we introduce an alternative formulation that allows us to\u0000establish GWP and scattering, even in the presence of poor time continuity of\u0000the Duhamel term.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260723","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
On singular behaviour in a plane linear elastostatics problem 论平面线性弹性力学问题中的奇异行为
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-12 DOI: arxiv-2409.07954
Heiko Gimperlein, Michael Grinfeld, Robin J. Knops, Marshall Slemrod
{"title":"On singular behaviour in a plane linear elastostatics problem","authors":"Heiko Gimperlein, Michael Grinfeld, Robin J. Knops, Marshall Slemrod","doi":"arxiv-2409.07954","DOIUrl":"https://doi.org/arxiv-2409.07954","url":null,"abstract":"A vector field similar to those separately introduced by Artstein and\u0000Dafermos is constructed from the tangent to a monotone increasing one-parameter\u0000family of non-concentric circles that touch at the common point of intersection\u0000taken as the origin. The circles define and space-fill a lens shaped region\u0000$Omega$ whose outer and inner boundaries are the greatest and least circles.\u0000The double cusp at the origin creates a geometric singularity at which the\u0000vector field is indeterminate and has non-unique limiting behaviour. A\u0000semi-inverse method that involves the Airy stress function then shows that the\u0000vector field corresponds to the displacement vector field for a linear plane\u0000compressible non-homogeneous isotropic elastostatic equilibrium problem in\u0000$Omega$ whose boundaries are rigidly rotated relative to each other, possibly\u0000causing rupture or tearing at the origin. A sequence of solutions is found for\u0000which not only are the Lam'{e} parameters strongly-elliptic, but the\u0000non-unique limiting behaviour of the displacement is preserved. Other\u0000properties of the vector field are also established.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179013","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
Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations with the Hall effect 具有霍尔效应的广义纳维-斯托克斯-麦克斯韦方程的良好拟合、磁螺旋守恒、不粘性极限和渐近稳定性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-12 DOI: arxiv-2409.07802
Kyungkeun Kang, Jihoon Lee, Dinh Duong Nguyen
{"title":"Well-posedness, magnetic helicity conservation, inviscid limit and asymptotic stability for the generalized Navier-Stokes-Maxwell equations with the Hall effect","authors":"Kyungkeun Kang, Jihoon Lee, Dinh Duong Nguyen","doi":"arxiv-2409.07802","DOIUrl":"https://doi.org/arxiv-2409.07802","url":null,"abstract":"This paper is devoted to studying the well-posedness, (conditional)\u0000conservation of magnetic helicity, inviscid limit and asymptotic stability of\u0000the generalized Navier-Stokes-Maxwell equations (NSM) under the Hall effect in\u0000two and three dimensions. More precisely, in the viscous case we prove the\u0000global well-posedness of NSM for small initial data, which allows us to\u0000establish a connection with either the Hall-magnetohydrodynamics (H-MHD) system\u0000as the speed of light tends to infinity or NSM without the Hall coefficient as\u0000this constant goes to zero. In addition, in the inviscid case the local\u0000well-posedness of NSM is also obtained for possibly large initial data.\u0000Moreover, under suitable conditions on the initial data and additional\u0000assumptions of solutions to NSM in three dimensions, the magnetic helicity is\u0000conserved as the electric conductivity goes to infinity. It is different to the\u0000case of the fractional H-MHD with critical fractional Laplacian exponents for\u0000both the velocity and magnetic fields, where the conservation of magnetic\u0000helicity can be provided for smooth initial data without any further conditions\u0000on the solution. Furthermore, the asymptotic stability of NSM around a constant\u0000magnetic field is established in the case of having a velocity damping term.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"61 28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179015","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
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