In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.
{"title":"Principal Frequency of $p$-Sub-Laplacians for General Vector Fields","authors":"Michael Ruzhansky, Bolys Sabitbek, D. Suragan","doi":"10.4171/ZAA/1674","DOIUrl":"https://doi.org/10.4171/ZAA/1674","url":null,"abstract":"In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73911508","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 : 2020-08-12DOI: 10.1142/s0219891620500113
P. Secchi
We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, in [10] the authors have derived a pseudo-differential equation which describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[vcdottau]|>2sqrt{2},c$, and the well-posedness was obtained in standard weighted Sobolev spaces. The aim of the present paper is to improve the result of [10], by showing the existence of the solution in function spaces with some additional weighted anisotropic regularity in the frequency space.
{"title":"Anisotropic regularity of linearized compressible vortex sheets","authors":"P. Secchi","doi":"10.1142/s0219891620500113","DOIUrl":"https://doi.org/10.1142/s0219891620500113","url":null,"abstract":"We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, in [10] the authors have derived a pseudo-differential equation which describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[vcdottau]|>2sqrt{2},c$, and the well-posedness was obtained in standard weighted Sobolev spaces. \u0000The aim of the present paper is to improve the result of [10], by showing the existence of the solution in function spaces with some additional weighted anisotropic regularity in the frequency space.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74287746","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}
K. Datchev, Jason Metcalfe, Jacob Shapiro, M. Tohaneanu
By considering a two ended warped product manifold, we demonstrate a bifurcation that can occur when metric trapping interacts with a boundary. In this highly symmetric example, as the boundary passes through the trapped set, one goes from a nontrapping scenario where lossless local energy estimates are available for the wave equation to the case of stably trapped rays where all but a logarithmic amount of decay is lost.
{"title":"On the interaction of metric trapping and a boundary","authors":"K. Datchev, Jason Metcalfe, Jacob Shapiro, M. Tohaneanu","doi":"10.1090/PROC/15460","DOIUrl":"https://doi.org/10.1090/PROC/15460","url":null,"abstract":"By considering a two ended warped product manifold, we demonstrate a bifurcation that can occur when metric trapping interacts with a boundary. In this highly symmetric example, as the boundary passes through the trapped set, one goes from a nontrapping scenario where lossless local energy estimates are available for the wave equation to the case of stably trapped rays where all but a logarithmic amount of decay is lost.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89577297","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 study the spatially homogeneous relativistic Boltzmann equation for massless particles in an FLRW background with scattering kernels in a certain range of soft and hard potentials. We obtain the future global existence of small solutions in a weighted $L^1cap L^infty$ space.
{"title":"The spatially homogeneous Boltzmann equation for massless particles in an FLRW background","authors":"Ho Lee","doi":"10.1063/5.0037951","DOIUrl":"https://doi.org/10.1063/5.0037951","url":null,"abstract":"We study the spatially homogeneous relativistic Boltzmann equation for massless particles in an FLRW background with scattering kernels in a certain range of soft and hard potentials. We obtain the future global existence of small solutions in a weighted $L^1cap L^infty$ space.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90202120","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 : 2020-08-06DOI: 10.1016/j.na.2020.112200
I. Fukuda, Kenta Itasaka
{"title":"Higher-order asymptotic profiles of the solutions to the viscous Fornberg–Whitham equation","authors":"I. Fukuda, Kenta Itasaka","doi":"10.1016/j.na.2020.112200","DOIUrl":"https://doi.org/10.1016/j.na.2020.112200","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88221763","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 announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are bounded above and below by flat horizontal walls and acted upon by gravity. We obtain continuous curves of solutions to this system that bifurcate from the trivial solution where the interface is flat. Following these families to the their extreme, the internal interface either overturns, comes into contact with the upper wall, or develops a highly degenerate "double stagnation" point. Our construction is made possible by a new abstract machinery for global continuation of monotone front-type solutions to elliptic equations posed on infinite cylinders. This theory is quite robust and, in particular, can treat fully nonlinear equations as well as quasilinear problems with transmission boundary conditions.
{"title":"Large-amplitude internal fronts in two-fluid systems","authors":"R. Chen, Samuel Walsh, Miles H. Wheeler","doi":"10.5802/crmath.128","DOIUrl":"https://doi.org/10.5802/crmath.128","url":null,"abstract":"In this announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are bounded above and below by flat horizontal walls and acted upon by gravity. We obtain continuous curves of solutions to this system that bifurcate from the trivial solution where the interface is flat. Following these families to the their extreme, the internal interface either overturns, comes into contact with the upper wall, or develops a highly degenerate \"double stagnation\" point. \u0000Our construction is made possible by a new abstract machinery for global continuation of monotone front-type solutions to elliptic equations posed on infinite cylinders. This theory is quite robust and, in particular, can treat fully nonlinear equations as well as quasilinear problems with transmission boundary conditions.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74306916","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 : 2020-07-27DOI: 10.1016/J.NA.2021.112494
Jincheng Gao, Zhengzhen Wei, Z. Yao
{"title":"Decay of Strong Solution for the Compressible Navier-Stokes Equations with Large Initial Data","authors":"Jincheng Gao, Zhengzhen Wei, Z. Yao","doi":"10.1016/J.NA.2021.112494","DOIUrl":"https://doi.org/10.1016/J.NA.2021.112494","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"532 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77701409","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 : 2020-07-25DOI: 10.4310/DPDE.2021.v18.n2.a3
D. Cao, Wei Dai, Yang Zhang
In this paper, we consider the following 2-D Schr"{o}dinger-Newton equations begin{eqnarray*} -Delta u+a(x)u+frac{gamma}{2pi}left(log(|cdot|)*|u|^pright){|u|}^{p-2}u=b{|u|}^{q-2}u qquad text{in} ,,, mathbb{R}^{2}, end{eqnarray*} where $ain C(mathbb{R}^{2})$ is a $mathbb{Z}^{2}$-periodic function with $inf_{mathbb{R}^{2}}a>0$, $gamma>0$, $bgeq0$, $pgeq2$ and $qgeq 2$. By using ideas from cite{CW,DW,Stubbe}, under mild assumptions, we obtain existence of ground state solutions and mountain pass solutions to the above equations for $pgeq2$ and $qgeq2p-2$ via variational methods. The auxiliary functional $J_{1}$ plays a key role in the cases $pgeq3$. We also prove the radial symmetry of positive solutions (up to translations) for $pgeq2$ and $qgeq 2$. The corresponding results for planar Schr"{o}dinger-Poisson systems will also be obtained. Our theorems extend the results in cite{CW,DW} from $p=2$ and $b=1$ to general $pgeq2$ and $bgeq0$.
{"title":"Existence and symmetry of solutions to 2-D Schrödinger–Newton equations","authors":"D. Cao, Wei Dai, Yang Zhang","doi":"10.4310/DPDE.2021.v18.n2.a3","DOIUrl":"https://doi.org/10.4310/DPDE.2021.v18.n2.a3","url":null,"abstract":"In this paper, we consider the following 2-D Schr\"{o}dinger-Newton equations begin{eqnarray*} -Delta u+a(x)u+frac{gamma}{2pi}left(log(|cdot|)*|u|^pright){|u|}^{p-2}u=b{|u|}^{q-2}u qquad text{in} ,,, mathbb{R}^{2}, end{eqnarray*} where $ain C(mathbb{R}^{2})$ is a $mathbb{Z}^{2}$-periodic function with $inf_{mathbb{R}^{2}}a>0$, $gamma>0$, $bgeq0$, $pgeq2$ and $qgeq 2$. By using ideas from cite{CW,DW,Stubbe}, under mild assumptions, we obtain existence of ground state solutions and mountain pass solutions to the above equations for $pgeq2$ and $qgeq2p-2$ via variational methods. The auxiliary functional $J_{1}$ plays a key role in the cases $pgeq3$. We also prove the radial symmetry of positive solutions (up to translations) for $pgeq2$ and $qgeq 2$. The corresponding results for planar Schr\"{o}dinger-Poisson systems will also be obtained. Our theorems extend the results in cite{CW,DW} from $p=2$ and $b=1$ to general $pgeq2$ and $bgeq0$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89760044","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 : 2020-07-24DOI: 10.4310/DPDE.2021.V18.N1.A5
Zhouyu Li, D. Zhou
Let $(u, pi)$ with $u=(u_1,u_2,u_3)$ be a suitable weak solution of the three dimensional Navier-Stokes equations in $mathbb{R}^3times [0, T]$. Denote by $dot{mathcal{B}}^{-1}_{infty,infty}$ the closure of $C_0^infty$ in $dot{B}^{-1}_{infty,infty}$. We prove that if $uin L^infty(0, T; dot{B}^{-1}_{infty,infty})$, $u(x, T)in dot{mathcal{B}}^{-1}_{infty,infty})$, and $u_3in L^infty(0, T; L^{3, infty})$ or $u_3in L^infty(0, T; dot{B}^{-1+3/p}_{p, q})$ with $3
{"title":"On endpoint regularity criterion of the 3D Navier–Stokes equations","authors":"Zhouyu Li, D. Zhou","doi":"10.4310/DPDE.2021.V18.N1.A5","DOIUrl":"https://doi.org/10.4310/DPDE.2021.V18.N1.A5","url":null,"abstract":"Let $(u, pi)$ with $u=(u_1,u_2,u_3)$ be a suitable weak solution of the three dimensional Navier-Stokes equations in $mathbb{R}^3times [0, T]$. Denote by $dot{mathcal{B}}^{-1}_{infty,infty}$ the closure of $C_0^infty$ in $dot{B}^{-1}_{infty,infty}$. We prove that if $uin L^infty(0, T; dot{B}^{-1}_{infty,infty})$, $u(x, T)in dot{mathcal{B}}^{-1}_{infty,infty})$, and $u_3in L^infty(0, T; L^{3, infty})$ or $u_3in L^infty(0, T; dot{B}^{-1+3/p}_{p, q})$ with $3<p, q< infty$, then $u$ is smooth in $mathbb{R}^3times [0, T]$. Our result improves a previous result established by Wang and Zhang [Sci. China Math. 60, 637-650 (2017)].","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"296 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77319122","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 obtain a genuine local $C^2$ estimate for the Monge-Ampere equation in dimension two, by using the partial Legendre transform.
利用偏勒让德变换，我们得到了二维蒙日-安培方程的一个真实的局部$C^2$估计。
{"title":"Interior $C^2$ estimate for Monge-Ampère equation in dimension two","authors":"Jiakun Liu","doi":"10.1090/PROC/15459","DOIUrl":"https://doi.org/10.1090/PROC/15459","url":null,"abstract":"We obtain a genuine local $C^2$ estimate for the Monge-Ampere equation in dimension two, by using the partial Legendre transform.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87938785","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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