Pub Date : 2020-11-08DOI: 10.1016/J.MATPUR.2021.01.008
Naian Liao
{"title":"Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior","authors":"Naian Liao","doi":"10.1016/J.MATPUR.2021.01.008","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.01.008","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77900331","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}
Pub Date : 2020-11-06DOI: 10.52843/meta-mat.c4qfhg
H. Ammari, B. Davies, Erik Orvehed Hiltunen, Hyundae Lee, Sanghyeon Yu
The aim of this review is to cover recent developments in the mathematical analysis of subwavelength resonators. The use of sophisticated mathematics in the field of metamaterials is reported, which provides a mathematical framework for focusing, trapping, and guiding of waves at subwavelength scales. Throughout this review, the power of layer potential techniques combined with asymptotic analysis for solving challenging wave propagation problems at subwavelength scales is demonstrated.
{"title":"Wave Interaction with Subwavelength Resonators","authors":"H. Ammari, B. Davies, Erik Orvehed Hiltunen, Hyundae Lee, Sanghyeon Yu","doi":"10.52843/meta-mat.c4qfhg","DOIUrl":"https://doi.org/10.52843/meta-mat.c4qfhg","url":null,"abstract":"The aim of this review is to cover recent developments in the mathematical analysis of subwavelength resonators. The use of sophisticated mathematics in the field of metamaterials is reported, which provides a mathematical framework for focusing, trapping, and guiding of waves at subwavelength scales. Throughout this review, the power of layer potential techniques combined with asymptotic analysis for solving challenging wave propagation problems at subwavelength scales is demonstrated.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83177903","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}
Pub Date : 2020-11-06DOI: 10.21494/iste.op.2021.0698
M. Hamouda, M. Hamza
We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) hspace{1cm} u_{tt}-t^{2m}Delta u=|u_t|^p+|u|^q, quad mbox{in} mathbb{R}^Ntimes[0,infty),$$ with small initial data, where $mge0$. For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $(Tr)$ with $mge0$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $m$. As an application of the method developed for the study of the equation $(Tr)$ we obtain with a different approach the same blow-up result as in cite{Lai2020} when we consider only one time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand side of $(Tr)$.
{"title":"Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities","authors":"M. Hamouda, M. Hamza","doi":"10.21494/iste.op.2021.0698","DOIUrl":"https://doi.org/10.21494/iste.op.2021.0698","url":null,"abstract":"We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) hspace{1cm} u_{tt}-t^{2m}Delta u=|u_t|^p+|u|^q, quad mbox{in} mathbb{R}^Ntimes[0,infty),$$ with small initial data, where $mge0$. For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $(Tr)$ with $mge0$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $m$. As an application of the method developed for the study of the equation $(Tr)$ we obtain with a different approach the same blow-up result as in cite{Lai2020} when we consider only one time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand side of $(Tr)$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86294608","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}
In this article we derive the expression of textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in (mathbb{R}^2) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-Helein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.
{"title":"Renormalized energies for unit-valued harmonic maps in multiply connected domains","authors":"Rémy Rodiac, Pa'ul Ubill'us","doi":"10.3233/ASY-211712","DOIUrl":"https://doi.org/10.3233/ASY-211712","url":null,"abstract":"In this article we derive the expression of textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in (mathbb{R}^2) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-Helein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85822970","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}
We review some recent results obtained for the time evolution of wave packets for systems of equations of pseudo-differential type, including Schr{"o}dinger ones, and discuss their application to the approximation of the associated unitary propagator. We start with scalar equations, propagation of coherent states, and applications to the Herman-Kluk approximation. Then we discuss the extension of these results to systems with eigenvalues of constant multiplicity or with smooth crossings.
{"title":"Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations","authors":"C. Kammerer, C. Lasser, D. Robert","doi":"10.4171/ecr/18-1/6","DOIUrl":"https://doi.org/10.4171/ecr/18-1/6","url":null,"abstract":"We review some recent results obtained for the time evolution of wave packets for systems of equations of pseudo-differential type, including Schr{\"o}dinger ones, and discuss their application to the approximation of the associated unitary propagator. We start with scalar equations, propagation of coherent states, and applications to the Herman-Kluk approximation. Then we discuss the extension of these results to systems with eigenvalues of constant multiplicity or with smooth crossings.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73581313","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}
We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation begin{align*} partial_{x}left(u_{t}-betapartial_{x}^{3}u +partial_{x}(u^{2})right)+partial_{y}^{2}u-gamma u=0 end{align*} in the anisotropic Sobolev spaces $H^{s_{1},>s_{2}}(mathbb{R}^{2})$. When $beta 0,$ we prove that the Cauchy problem is locally well-posed in $H^{s_{1},>s_{2}}(mathbb{R}^{2})$ with $s_{1}>-frac{1}{2}$ and $s_{2}geq 0$. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang( Transactions of the American Mathematical Society, 364(2012), 3395--3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in $H^{s_{1},>0}(mathbb{R}^{2})$ with $s_{1} 0,$ by using the $U^{p}$ and $V^{p}$ spaces, we prove that the Cauchy problem is locally well-posed in $H^{-frac{1}{2},>0}(mathbb{R}^{2})$.
{"title":"Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces","authors":"Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan","doi":"10.3934/dcds.2021097","DOIUrl":"https://doi.org/10.3934/dcds.2021097","url":null,"abstract":"We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation begin{align*} partial_{x}left(u_{t}-betapartial_{x}^{3}u +partial_{x}(u^{2})right)+partial_{y}^{2}u-gamma u=0 end{align*} in the anisotropic Sobolev spaces $H^{s_{1},>s_{2}}(mathbb{R}^{2})$. When $beta 0,$ we prove that the Cauchy problem is locally well-posed in $H^{s_{1},>s_{2}}(mathbb{R}^{2})$ with $s_{1}>-frac{1}{2}$ and $s_{2}geq 0$. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang( Transactions of the American Mathematical Society, 364(2012), 3395--3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in $H^{s_{1},>0}(mathbb{R}^{2})$ with $s_{1} 0,$ by using the $U^{p}$ and $V^{p}$ spaces, we prove that the Cauchy problem is locally well-posed in $H^{-frac{1}{2},>0}(mathbb{R}^{2})$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83177827","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}
We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems of coupled quantum harmonic oscillators with arbitrary frequencies. The structure of the set of semi-classical measures turns out to depend strongly on the arithmetic relations between frequencies of each decoupled oscillator. In particular, we show that as soon as these frequencies are not rational multiples of a fixed fundamental frequency, the set of semi-classical measures is not convex and therefore, infinitely many measures that are invariant under the classical harmonic oscillator are not semi-classical measures.
{"title":"Localization and delocalization of eigenmodes of Harmonic oscillators","authors":"V'ictor Arnaiz, F. Macià","doi":"10.1090/proc/15767","DOIUrl":"https://doi.org/10.1090/proc/15767","url":null,"abstract":"We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems of coupled quantum harmonic oscillators with arbitrary frequencies. The structure of the set of semi-classical measures turns out to depend strongly on the arithmetic relations between frequencies of each decoupled oscillator. In particular, we show that as soon as these frequencies are not rational multiples of a fixed fundamental frequency, the set of semi-classical measures is not convex and therefore, infinitely many measures that are invariant under the classical harmonic oscillator are not semi-classical measures.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74774439","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}
Pub Date : 2020-10-26DOI: 10.1016/J.NA.2021.112362
H'ector A. Chang-Lara, Edgard A. Pimentel
{"title":"Non-convex Hamilton-Jacobi equations with gradient constraints","authors":"H'ector A. Chang-Lara, Edgard A. Pimentel","doi":"10.1016/J.NA.2021.112362","DOIUrl":"https://doi.org/10.1016/J.NA.2021.112362","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83593920","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}
Pub Date : 2020-10-25DOI: 10.1016/J.NA.2021.112530
Chenmin Sun, N. Tzvetkov
{"title":"Refined probabilistic global well-posedness for the weakly dispersive NLS","authors":"Chenmin Sun, N. Tzvetkov","doi":"10.1016/J.NA.2021.112530","DOIUrl":"https://doi.org/10.1016/J.NA.2021.112530","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79975898","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}
In this paper, we consider a doubly nonlinear parabolic equation $ partial _t beta (u) - nabla cdot alpha (x , nabla u) ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $beta : mathbb{R} to 2 ^{ mathbb{R} }$ is a maximal monotone graph satisfying $0 in beta (0)$ and $ nabla cdot alpha (x , nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $beta $. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for $1 < p < 2$. Main purpose of this paper is to show the solvability of the initial boundary value problem for any $ p in (1, infty ) $ without any conditions for $beta $ except $0 in beta (0)$. We also discuss the uniqueness of solution by using properties of entropy solution.
本文考虑一类双非线性抛物型方程 $ partial _t beta (u) - nabla cdot alpha (x , nabla u) ni f$ 具有有界域上齐次Dirichlet边界条件，其中 $beta : mathbb{R} to 2 ^{ mathbb{R} }$ 极大单调图是否令人满意 $0 in beta (0)$ 和 $ nabla cdot alpha (x , nabla u )$ 代表广义的 $p$——拉普拉斯。在单值性、强制性或某些生长条件下，对该方程初边值问题解的存在性进行了大量的研究 $beta $． 然而，有一些结果的情况下，这些假设被删除，这是很难构建一个抽象的理论涵盖的情况下 $1 < p < 2$． 本文的主要目的是证明任意方程的初边值问题的可解性 $ p in (1, infty ) $ 没有任何条件 $beta $ 除了 $0 in beta (0)$． 并利用熵解的性质讨论了解的唯一性。
{"title":"Solvability of doubly nonlinear parabolic equation with p-laplacian","authors":"S. Uchida","doi":"10.3934/eect.2021033","DOIUrl":"https://doi.org/10.3934/eect.2021033","url":null,"abstract":"In this paper, we consider a doubly nonlinear parabolic equation $ partial _t beta (u) - nabla cdot alpha (x , nabla u) ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $beta : mathbb{R} to 2 ^{ mathbb{R} }$ is a maximal monotone graph satisfying $0 in beta (0)$ and $ nabla cdot alpha (x , nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $beta $. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for $1 < p < 2$. Main purpose of this paper is to show the solvability of the initial boundary value problem for any $ p in (1, infty ) $ without any conditions for $beta $ except $0 in beta (0)$. We also discuss the uniqueness of solution by using properties of entropy solution.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90935073","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}
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