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On the stochastic Allen–Cahn equation on networks with multiplicative noise 带乘性噪声网络上的随机Allen-Cahn方程
arXiv: Analysis of PDEs Pub Date : 2020-11-23 DOI: 10.14232/EJQTDE.2021.1.7
M. Kov'acs, E. Sikolya
{"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}
引用次数: 6
Travelling wave solutions for gravity fingering in porous media flows. 多孔介质流动中重力指法的行波解。
arXiv: Analysis of PDEs Pub Date : 2020-11-21 DOI: 10.13140/RG.2.2.23096.78083
K. Mitra, A. Ratz, B. Schweizer
{"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}
引用次数: 3
Dual variational methods for an indefinte nonlinear Helmholtz equation 一类不定非线性亥姆霍兹方程的对偶变分方法
arXiv: Analysis of PDEs Pub Date : 2020-11-16 DOI: 10.5445/IR/1000126434/V2
Rainer Mandel, Dominic Scheider, Tolga A Yeşil
{"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}
引用次数: 1
Potential well theory for the derivative nonlinearSchrödinger equation 势阱理论的导数nonlinearSchrödinger方程
arXiv: Analysis of PDEs Pub Date : 2020-11-16 DOI: 10.2140/apde.2021.14.909
M. Hayashi
{"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}
引用次数: 8
Spatial diffusion and periodic evolving of domain in an SIS epidemic model SIS流行病模型中区域的空间扩散和周期演化
arXiv: Analysis of PDEs Pub Date : 2020-11-15 DOI: 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}
引用次数: 2
Generalized Carleson perturbations of elliptic operators and applications 椭圆算子的广义Carleson摄动及其应用
arXiv: Analysis of PDEs Pub Date : 2020-11-12 DOI: 10.1090/tran/8635
J. Feneuil, Bruno Poggi
{"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}
引用次数: 9
Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity 组合非线性广义Tricomi方程整体解的不存在性
arXiv: Analysis of PDEs Pub Date : 2020-11-11 DOI: 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}
引用次数: 8
Entropy admissibility of the limit solution for a nonlocal model of traffic flow 一类非局部交通流模型极限解的熵容许性
arXiv: Analysis of PDEs Pub Date : 2020-11-10 DOI: 10.4310/cms.2021.v19.n5.a12
A. Bressan, Wen Shen
{"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}
引用次数: 25
On the existence and uniqueness of weak solutions to time-fractional elliptic equations with time-dependent variable coefficients 时变系数时间分数型椭圆方程弱解的存在唯一性
arXiv: Analysis of PDEs Pub Date : 2020-11-10 DOI: 10.1090/PROC/15533
H. T. Tuan
{"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}
引用次数: 1
Bergman-Bourgain-Brezis-type Inequality Bergman-Bourgain-Brezis-type不平等
arXiv: Analysis of PDEs Pub Date : 2020-11-08 DOI: 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}
引用次数: 2
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