Pub Date : 2021-01-01DOI: 10.4310/maa.2021.v28.n4.a7
Binbin Shi, Weike Wang
{"title":"Suppression of blowup by mixing in an aggregation equation with supercritical dissipation","authors":"Binbin Shi, Weike Wang","doi":"10.4310/maa.2021.v28.n4.a7","DOIUrl":"https://doi.org/10.4310/maa.2021.v28.n4.a7","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489459","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 : 2021-01-01DOI: 10.4310/maa.2021.v28.n3.a1
Yuehong Feng, Xin Li, Shu Wang
{"title":"Hsiao’s PDE theory on semi-conductor and plasma and their applications","authors":"Yuehong Feng, Xin Li, Shu Wang","doi":"10.4310/maa.2021.v28.n3.a1","DOIUrl":"https://doi.org/10.4310/maa.2021.v28.n3.a1","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70489243","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 : 2020-08-13DOI: 10.4310/MAA.2021.v28.n1.a1
E. Indrei
A weighted relative isoperimetric inequality in convex cones is obtained via the Monge-Ampere equation.
利用蒙日-安培方程得到凸锥的加权相对等周不等式。
{"title":"A weighted relative isoperimetric inequality in convex cones","authors":"E. Indrei","doi":"10.4310/MAA.2021.v28.n1.a1","DOIUrl":"https://doi.org/10.4310/MAA.2021.v28.n1.a1","url":null,"abstract":"A weighted relative isoperimetric inequality in convex cones is obtained via the Monge-Ampere equation.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48362119","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 : 2020-08-13DOI: 10.4310/maa.2022.v29.n3.a4
Billel Guelmame, D. Clamond, S. Junca
We prove in this note the local (in time) well-posedness of a broad class of $2 times 2$ symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond et al. [2]. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.
{"title":"Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom","authors":"Billel Guelmame, D. Clamond, S. Junca","doi":"10.4310/maa.2022.v29.n3.a4","DOIUrl":"https://doi.org/10.4310/maa.2022.v29.n3.a4","url":null,"abstract":"We prove in this note the local (in time) well-posedness of a broad class of $2 times 2$ symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond et al. [2]. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49089501","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 : 2020-07-16DOI: 10.4310/maa.2021.v28.n2.a3
Li Chen, Changhui Tan, Lining Tong
We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.
{"title":"On the global classical solution to compressible Euler system with singular velocity alignment","authors":"Li Chen, Changhui Tan, Lining Tong","doi":"10.4310/maa.2021.v28.n2.a3","DOIUrl":"https://doi.org/10.4310/maa.2021.v28.n2.a3","url":null,"abstract":"We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49667477","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 : 2020-07-06DOI: 10.4310/maa.2021.v28.n1.a4
G. Loeper, N. Trudinger
We simplify the geometric interpretation of the weak Ma-Trudinger-Wang condition for regularity in optimal transportation and provide a geometric proof of the global c-convexity of locally $c$-convex potentials when the cost function $c$ is only assumed twice differentiable.
{"title":"Weak formulation of the MTW condition and convexity properties of potentials","authors":"G. Loeper, N. Trudinger","doi":"10.4310/maa.2021.v28.n1.a4","DOIUrl":"https://doi.org/10.4310/maa.2021.v28.n1.a4","url":null,"abstract":"We simplify the geometric interpretation of the weak Ma-Trudinger-Wang condition for regularity in optimal transportation and provide a geometric proof of the global c-convexity of locally $c$-convex potentials when the cost function $c$ is only assumed twice differentiable.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47585862","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 : 2020-05-27DOI: 10.4310/maa.2020.v27.n4.a5
Nestor Guillen, J. Kitagawa
In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second boundary value problem. Our main observation is that this problem can be posed as an optimal transport problem where the target is a subset of the lower hemisphere of $mathbb{S}^n$. As a result we obtain existence and regularity of solutions under mild assumptions on the curvature, as well as a quantitative version of a gradient blowup result due to Urbas, which turns out to fall within the optimal transport framework.
{"title":"Optimal transport and the Gauss curvature equation","authors":"Nestor Guillen, J. Kitagawa","doi":"10.4310/maa.2020.v27.n4.a5","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n4.a5","url":null,"abstract":"In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second boundary value problem. Our main observation is that this problem can be posed as an optimal transport problem where the target is a subset of the lower hemisphere of $mathbb{S}^n$. As a result we obtain existence and regularity of solutions under mild assumptions on the curvature, as well as a quantitative version of a gradient blowup result due to Urbas, which turns out to fall within the optimal transport framework.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42687731","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 : 2020-03-24DOI: 10.4310/maa.2022.v29.n1.a1
W. Strauss, Masahiro Suzuki
We consider the steady states of a gas between two parallel plates that is ionized by a strong electric field so as to create a plasma. There can be a cascade of electrons due both to the electrons colliding with the gas molecules and to the ions colliding with the cathode (secondary emission). We use global bifurcation theory to prove that there is a one-parameter family $mathscr{K}$ of such steady states with the following property. The curve $mathscr{K}$ begins at the sparking voltage and either the particle density becomes unbounded or $mathscr{K}$ ends at an anti-sparking voltage. These critical voltages are characterized explicitly.
{"title":"Steady states of gas ionization with secondary emission","authors":"W. Strauss, Masahiro Suzuki","doi":"10.4310/maa.2022.v29.n1.a1","DOIUrl":"https://doi.org/10.4310/maa.2022.v29.n1.a1","url":null,"abstract":"We consider the steady states of a gas between two parallel plates that is ionized by a strong electric field so as to create a plasma. There can be a cascade of electrons due both to the electrons colliding with the gas molecules and to the ions colliding with the cathode (secondary emission). We use global bifurcation theory to prove that there is a one-parameter family $mathscr{K}$ of such steady states with the following property. The curve $mathscr{K}$ begins at the sparking voltage and either the particle density becomes unbounded or $mathscr{K}$ ends at an anti-sparking voltage. These critical voltages are characterized explicitly.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43641095","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 : 2020-03-24DOI: 10.4310/maa.2021.v28.n1.a5
Brendan Pass, A. Pinamonti, Mattia Vedovato
We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot-Caratheodory distances from the marginal points to their barycenter. Under certain technical hypotheses, we prove existence and uniqueness of optimal maps. We also point out several related open questions.
{"title":"Multi-marginal optimal transport on the Heisenberg group","authors":"Brendan Pass, A. Pinamonti, Mattia Vedovato","doi":"10.4310/maa.2021.v28.n1.a5","DOIUrl":"https://doi.org/10.4310/maa.2021.v28.n1.a5","url":null,"abstract":"We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot-Caratheodory distances from the marginal points to their barycenter. Under certain technical hypotheses, we prove existence and uniqueness of optimal maps. We also point out several related open questions.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41621518","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 : 2020-01-07DOI: 10.4310/maa.2021.v28.n1.a2
N. Le
In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge-Amp`ere equations begin{equation*} left{ begin{alignedat}{2} det D^2 u~& = gamma |v|^p~&&text{in} ~ Omega, det D^2 v~& = mu |u|^{n^2/p}~&&text{in} ~ Omega, u=v &= 0~&&text{on}~ partialOmega end{alignedat} right. end{equation*} on bounded, smooth and uniformly convex domains $Omegasubset R^n$ provided that $p$ is close to $ngeq 2$. When $p=n$, we show that the uniqueness holds for general bounded convex domains $Omegasubset R^n$.
{"title":"Uniqueness for a system of Monge–Ampère equations","authors":"N. Le","doi":"10.4310/maa.2021.v28.n1.a2","DOIUrl":"https://doi.org/10.4310/maa.2021.v28.n1.a2","url":null,"abstract":"In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge-Amp`ere equations begin{equation*} left{ begin{alignedat}{2} det D^2 u~& = gamma |v|^p~&&text{in} ~ Omega, det D^2 v~& = mu |u|^{n^2/p}~&&text{in} ~ Omega, u=v &= 0~&&text{on}~ partialOmega end{alignedat} right. end{equation*} on bounded, smooth and uniformly convex domains $Omegasubset R^n$ provided that $p$ is close to $ngeq 2$. When $p=n$, we show that the uniqueness holds for general bounded convex domains $Omegasubset R^n$.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47723999","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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