Pub Date : 2020-11-23DOI: 10.14232/EJQTDE.2021.1.7
M. Kov'acs, E. Sikolya
We consider a system of stochastic Allen-Cahn equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative Gaussian noise driven stochastic Allen-Cahn equation is given with possibly different potential barrier heights supplemented by a continuity condition and a Kirchhoff-type law in the vertices. Using the semigroup approach for stochastic evolution equations in Banach spaces we obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. We also prove more precise space-time regularity of the solution.
{"title":"On the stochastic Allen–Cahn equation on networks with multiplicative noise","authors":"M. Kov'acs, E. Sikolya","doi":"10.14232/EJQTDE.2021.1.7","DOIUrl":"https://doi.org/10.14232/EJQTDE.2021.1.7","url":null,"abstract":"We consider a system of stochastic Allen-Cahn equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative Gaussian noise driven stochastic Allen-Cahn equation is given with possibly different potential barrier heights supplemented by a continuity condition and a Kirchhoff-type law in the vertices. Using the semigroup approach for stochastic evolution equations in Banach spaces we obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. We also prove more precise space-time regularity of the solution.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73238226","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 : 2020-11-21DOI: 10.13140/RG.2.2.23096.78083
K. Mitra, A. Ratz, B. Schweizer
We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurrence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper, we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.
{"title":"Travelling wave solutions for gravity fingering in porous media flows.","authors":"K. Mitra, A. Ratz, B. Schweizer","doi":"10.13140/RG.2.2.23096.78083","DOIUrl":"https://doi.org/10.13140/RG.2.2.23096.78083","url":null,"abstract":"We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurrence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper, we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77385862","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 : 2020-11-16DOI: 10.5445/IR/1000126434/V2
Rainer Mandel, Dominic Scheider, Tolga A Yeşil
We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$-Delta u-k^2u=Q(x)|u|^{p-2}u,quad uin W^{2,p}(mathbb{R}^N)$$ with $k>0, Nge3,pinleft[frac{2(N+1)}{N-1},frac{2N}{N-2}right]$ and $Qin L^infty(mathbb{R}^N)$. Due to sign-changes of $Q$, our solutions have infinite Morse-Index in the corresponding dual variational formulation.
{"title":"Dual variational methods for an indefinte nonlinear Helmholtz equation","authors":"Rainer Mandel, Dominic Scheider, Tolga A Yeşil","doi":"10.5445/IR/1000126434/V2","DOIUrl":"https://doi.org/10.5445/IR/1000126434/V2","url":null,"abstract":"We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$-Delta u-k^2u=Q(x)|u|^{p-2}u,quad uin W^{2,p}(mathbb{R}^N)$$ with $k>0, Nge3,pinleft[frac{2(N+1)}{N-1},frac{2N}{N-2}right]$ and $Qin L^infty(mathbb{R}^N)$. Due to sign-changes of $Q$, our solutions have infinite Morse-Index in the \u0000corresponding dual variational formulation.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77061729","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 : 2020-11-16DOI: 10.2140/apde.2021.14.909
M. Hayashi
We consider the following nonlinear Schrodinger equation of derivative type: begin{equation}i partial_t u + partial_x^2 u +i |u|^{2} partial_x u +b|u|^4u=0 , quad (t,x) in mathbb{R}timesmathbb{R}, b inmathbb{R}. end{equation} If $b=0$, this equation is known as a gauge equivalent form of well-known derivative nonlinear Schrodinger equation (DNLS), which is mass critical and completely integrable. The equation can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data $u_0in H^1(mathbb{R})$ satisfies the mass condition $| u_0|_{L^2}^2 <4pi$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation for general $binmathbb{R}$, which is exactly corresponding to $4pi$-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both $4pi$-mass condition and algebraic solitons.
我们考虑以下导数型非线性薛定谔方程:begin{equation}i partial_t u + partial_x^2 u +i |u|^{2} partial_x u +b|u|^4u=0 , quad (t,x) in mathbb{R}timesmathbb{R}, b inmathbb{R}. end{equation}如果$b=0$，该方程被称为众所周知的导数型非线性薛定谔方程(DNLS)的规范等价形式，它是质量临界的，完全可积的。在保持质量临界性和哈密顿结构的前提下，该方程可视为DNLS的广义方程。对于DNLS，已知如果初始数据$u_0in H^1(mathbb{R})$满足质量条件$| u_0|_{L^2}^2 <4pi$，其解是全局有界的。本文首先在一般的$binmathbb{R}$方程上建立了与DNLS方程$4pi$ -质量条件完全对应的质量条件，然后从势阱理论的角度对其进行了表征。我们看到，质量阈值给出了由孤子产生的势阱结构的转折点。特别地，我们的结果给出了$4pi$ -质量条件和代数孤子的特征。
{"title":"Potential well theory for the derivative nonlinear\u0000Schrödinger equation","authors":"M. Hayashi","doi":"10.2140/apde.2021.14.909","DOIUrl":"https://doi.org/10.2140/apde.2021.14.909","url":null,"abstract":"We consider the following nonlinear Schrodinger equation of derivative type: begin{equation}i partial_t u + partial_x^2 u +i |u|^{2} partial_x u +b|u|^4u=0 , quad (t,x) in mathbb{R}timesmathbb{R}, b inmathbb{R}. end{equation} If $b=0$, this equation is known as a gauge equivalent form of well-known derivative nonlinear Schrodinger equation (DNLS), which is mass critical and completely integrable. The equation can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data $u_0in H^1(mathbb{R})$ satisfies the mass condition $| u_0|_{L^2}^2 <4pi$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation for general $binmathbb{R}$, which is exactly corresponding to $4pi$-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both $4pi$-mass condition and algebraic solitons.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79004513","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 : 2020-11-15DOI: 10.1016/J.NONRWA.2021.103343
Yachun Tong, Zhigui Lin
{"title":"Spatial diffusion and periodic evolving of domain in an SIS epidemic model","authors":"Yachun Tong, Zhigui Lin","doi":"10.1016/J.NONRWA.2021.103343","DOIUrl":"https://doi.org/10.1016/J.NONRWA.2021.103343","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74032836","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 extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators. Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of $A_{infty}$ among elliptic measures.
{"title":"Generalized Carleson perturbations of elliptic operators and applications","authors":"J. Feneuil, Bruno Poggi","doi":"10.1090/tran/8635","DOIUrl":"https://doi.org/10.1090/tran/8635","url":null,"abstract":"We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators. \u0000Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of $A_{infty}$ among elliptic measures.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90928370","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 : 2020-11-11DOI: 10.1016/j.nonrwa.2021.103354
Wenhui Chen, S. Lucente, A. Palmieri
{"title":"Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity","authors":"Wenhui Chen, S. Lucente, A. Palmieri","doi":"10.1016/j.nonrwa.2021.103354","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2021.103354","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75637214","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 : 2020-11-10DOI: 10.4310/cms.2021.v19.n5.a12
A. Bressan, Wen Shen
We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $rho$ ahead. The averaging kernel is of exponential type: $w_varepsilon(s)=varepsilon^{-1} e^{-s/varepsilon}$. For any decreasing velocity function $v$, we prove that, as $varepsilonto 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law $rho_t + (rho v(rho))_x=0$.
{"title":"Entropy admissibility of the limit solution for a nonlocal model of traffic flow","authors":"A. Bressan, Wen Shen","doi":"10.4310/cms.2021.v19.n5.a12","DOIUrl":"https://doi.org/10.4310/cms.2021.v19.n5.a12","url":null,"abstract":"We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $rho$ ahead. The averaging kernel is of exponential type: $w_varepsilon(s)=varepsilon^{-1} e^{-s/varepsilon}$. For any decreasing velocity function $v$, we prove that, as $varepsilonto 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law $rho_t + (rho v(rho))_x=0$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87341760","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}
This paper is devoted to discussing the existence and uniqueness of weak solutions to time-fractional elliptic equations having time-dependent variable coefficients. To obtain the main result, our strategy is to combine the Galerkin method, a basic inequality for the fractional derivative of convex Lyapunov candidate functions, the Yoshida approximation sequence and the weak compactness argument.
{"title":"On the existence and uniqueness of weak solutions to time-fractional elliptic equations with time-dependent variable coefficients","authors":"H. T. Tuan","doi":"10.1090/PROC/15533","DOIUrl":"https://doi.org/10.1090/PROC/15533","url":null,"abstract":"This paper is devoted to discussing the existence and uniqueness of weak solutions to time-fractional elliptic equations having time-dependent variable coefficients. To obtain the main result, our strategy is to combine the Galerkin method, a basic inequality for the fractional derivative of convex Lyapunov candidate functions, the Yoshida approximation sequence and the weak compactness argument.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90756059","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 : 2020-11-08DOI: 10.1016/j.jfa.2021.109201
F. Lio, T. Rivière, J. Wettstein
{"title":"Bergman-Bourgain-Brezis-type Inequality","authors":"F. Lio, T. Rivière, J. Wettstein","doi":"10.1016/j.jfa.2021.109201","DOIUrl":"https://doi.org/10.1016/j.jfa.2021.109201","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":"75109774","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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