Mathematics in Engineering最新文献_第5页

Fractional KPZ equations with fractional gradient term and Hardy potential 具有分数阶梯度项和Hardy势的分数阶KPZ方程

IF 1 4区工程技术

Mathematics in Engineering Pub Date : 2022-01-01 DOI: 10.3934/mine.2023042

B. Abdellaoui, K. Biroud, A. Primo, Fernando Soria, Abdelbadie Younes

{"title":"Fractional KPZ equations with fractional gradient term and Hardy potential","authors":"B. Abdellaoui, K. Biroud, A. Primo, Fernando Soria, Abdelbadie Younes","doi":"10.3934/mine.2023042","DOIUrl":"https://doi.org/10.3934/mine.2023042","url":null,"abstract":"In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem begin{document}$ left{ begin{array}{rcll} (-Delta )^s u & = &lambda dfrac{u}{|x|^{2s}}+ (mathfrak{F}(u)(x))^p+ rho f & text{ in } Omega, u&>&0 & text{ in }Omega, u& = &0 & text{ in }(mathbb{R}^NsetminusOmega), end{array}right. $end{document} where $ Omegasubset mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N > 2s, rho > 0 $, $ 0 < s < 1 $, $ 1 < p < infty $ and $ 0 < lambda < Lambda_{N, s} $, the Hardy constant defined below. We assume that $ f $ is a non-negative function with additional hypotheses. Here $ mathfrak{F}(u) $ is a nonlocal \"gradient\" term. In particular, if $ mathfrak{F}(u)(x) = |(-Delta)^{frac s2}u(x)| $, then we are able to show the existence of a critical exponents $ p_{+}(lambda, s) $ such that: 1) if $ p > p_{+}(lambda, s) $, there is no positive solution, 2) if $ p < p_{+}(lambda, s) $, there exists, at least, a positive supersolution solution for suitable data and $ rho $ small. Moreover, under additional restriction on $ p $, there exists a solution for general datum $ f $.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Universal potential estimates for $ 1 < pleq 2-frac{1}{n} $ 普遍潜在估计 $ 1 < pleq 2-frac{1}{n} $

IF 1 4区工程技术

Mathematics in Engineering Pub Date : 2022-01-01 DOI: 10.3934/mine.2023057

Quoc-Hung Nguyen, N. Phuc

引用次数: 1

Instabilities in internal gravity waves 内部重力波的不稳定性

IF 1 4区工程技术

Mathematics in Engineering Pub Date : 2022-01-01 DOI: 10.3934/mine.2023016

Dheeraj Varma, M. Mathur, T. Dauxois

{"title":"Instabilities in internal gravity waves","authors":"Dheeraj Varma, M. Mathur, T. Dauxois","doi":"10.3934/mine.2023016","DOIUrl":"https://doi.org/10.3934/mine.2023016","url":null,"abstract":"Internal gravity waves are propagating disturbances in stably stratified fluids, and can transport momentum and energy over large spatial extents. From a fundamental viewpoint, internal waves are interesting due to the nature of their dispersion relation, and their linear dynamics are reasonably well-understood. From an oceanographic viewpoint, a qualitative and quantitative understanding of significant internal wave generation in the ocean is emerging, while their dissipation mechanisms are being debated. This paper reviews the current knowledge on instabilities in internal gravity waves, primarily focusing on the growth of small-amplitude disturbances. Historically, wave-wave interactions based on weakly nonlinear expansions have driven progress in this field, to investigate spontaneous energy transfer to various temporal and spatial scales. Recent advances in numerical/experimental modeling and field observations have further revealed noticeable differences between various internal wave spatial forms in terms of their instability characteristics; this in turn has motivated theoretical calculations on appropriately chosen internal wave fields in various settings. After a brief introduction, we present a pedagogical discussion on linear internal waves and their different two-dimensional spatial forms. The general ideas concerning triadic resonance in internal waves are then introduced, before proceeding towards instability characteristics of plane waves, wave beams and modes. Results from various theoretical, experimental and numerical studies are summarized to provide an overall picture of the gaps in our understanding. An ocean perspective is then given, both in terms of the relevant outstanding questions and the various additional factors at play. While the applications in this review are focused on the ocean, several ideas are relevant to atmospheric and astrophysical systems too.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Some comparison results and a partial bang-bang property for two-phases problems in balls 球中两相问题的一些比较结果和部分bang-bang性质

IF 1 4区工程技术

Mathematics in Engineering Pub Date : 2022-01-01 DOI: 10.3934/mine.2023010

Idriss Mazari

引用次数: 2

Isoparametric foliations and the Pompeiu property 等参叶理和庞贝性质

IF 1 4区工程技术

Mathematics in Engineering Pub Date : 2022-01-01 DOI: 10.3934/mine.2023031

L. Provenzano, A. Savo

引用次数: 1

Qualitative properties of solutions to the Dirichlet problem for a Laplace equation involving the Hardy potential with possibly boundary singularity 含可能边界奇点的Hardy势拉普拉斯方程Dirichlet问题解的定性性质