{"title":"Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation","authors":"Weiwei Ao, J. Dávila, Manuel del Pino, M. Musso, Juncheng Wei","doi":"10.1090/TRAN/8406","DOIUrl":"https://doi.org/10.1090/TRAN/8406","url":null,"abstract":"For the generalized surface quasi-geostrophic equation $$left{ begin{aligned} & partial_t theta+ucdot nabla theta=0, quad text{in } mathbb{R}^2 times (0,T), & u=nabla^perp psi, quad psi = (-Delta)^{-s}theta quad text{in } mathbb{R}^2 times (0,T) , end{aligned} right. $$ $0<s<1$, we consider for $kge1$ the problem of finding a family of $k$-vortex solutions $theta_varepsilon(x,t)$ such that as $varepsilonto 0$ $$ theta_varepsilon(x,t) rightharpoonup sum_{j=1}^k m_jdelta(x-xi_j(t)) $$ for suitable trajectories for the vortices $x=xi_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-Delta)^sW = (W-1)^gamma_+, quad text{in } mathbb{R}^2, quad 1<gamma < frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in cite{chan_uniqueness_2020}.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87202572","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 : 2020-08-25DOI: 10.4310/CMS.2021.V19.N1.A10
Tong Li, Dehua Wang, Fang Wang, Zhian Wang, Kun Zhao
We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero.
{"title":"Large time behavior and diffusion limit for a system of balance laws from chemotaxis in multi-dimensions","authors":"Tong Li, Dehua Wang, Fang Wang, Zhian Wang, Kun Zhao","doi":"10.4310/CMS.2021.V19.N1.A10","DOIUrl":"https://doi.org/10.4310/CMS.2021.V19.N1.A10","url":null,"abstract":"We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90884967","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}
Given a smooth Hermitian vector bundle $mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $nabla^{mathcal{E}}$ on the vector bundle $mathcal{E}$. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when $dim(M) geq 3$, answering an open question of Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $nabla^{mathrm{End}(mathcal{E})}$ on the endomorphism bundle $mathrm{End}(mathcal{E})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the geodesic flow is Anosov on the unit tangent bundle), the connections are generically $textit{opaque}$, namely there are no non-trivial subbundles of $mathcal{E}$ which are generically preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called $textit{operators of uniform divergence type}$, and on perturbative arguments from spectral theory (especially on the theory of Pollicott-Ruelle resonances in the Anosov case).
给定闭黎曼流形$(M,g)$上的光滑厄米向量束$mathcal{E}$，研究了该向量束$mathcal{E}$上的幺正连接$nabla^{mathcal{E}}$的一般性质。首先，我们证明了扭曲保形杀伤张量(ckt)在$dim(M) geq 3$时是一般平凡的，回答了guillarmo - paternain - salo - uhlmann的一个开放问题。在负曲率下，已知扭曲ckt的存在是精确求解扭曲上同调方程的唯一障碍，而扭曲上同调方程可能出现在各种几何问题中，如透明连接的研究。本文的主要结果表明，这些方程是可以一般求解的。作为一个副产品，我们也得到了在自同态束$mathrm{End}(mathcal{E})$上的诱导连接$nabla^{mathrm{End}(mathcal{E})}$具有一般平凡的ckt，只要$(M,g)$在其平凡的线束上没有非平凡的ckt。最后，我们证明，在附加假设$(M,g)$是Anosov(即测地线流是单位切线束上的Anosov)的情况下，连接一般为$textit{opaque}$，即不存在$mathcal{E}$的非平凡子束，这些子束一般由沿测地线的平行传输保存。这些证明依赖于(伪)微分算子$textit{operators of uniform divergence type}$的一个新的微局部性质的引入，以及谱理论的微扰论证(特别是在Anosov情况下的pollicot - ruelle共振理论)。
{"title":"Generic Dynamical Properties of Connections on Vector Bundles","authors":"Mihajlo Ceki'c, Thibault Lefeuvre","doi":"10.1093/IMRN/RNAB069","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB069","url":null,"abstract":"Given a smooth Hermitian vector bundle $mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $nabla^{mathcal{E}}$ on the vector bundle $mathcal{E}$. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when $dim(M) geq 3$, answering an open question of Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $nabla^{mathrm{End}(mathcal{E})}$ on the endomorphism bundle $mathrm{End}(mathcal{E})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the geodesic flow is Anosov on the unit tangent bundle), the connections are generically $textit{opaque}$, namely there are no non-trivial subbundles of $mathcal{E}$ which are generically preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called $textit{operators of uniform divergence type}$, and on perturbative arguments from spectral theory (especially on the theory of Pollicott-Ruelle resonances in the Anosov case).","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74418109","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 investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: begin{equation*} -left( aint_{mathbb{R}^{N}}|nabla u|^{2}dx+1right) Delta u+mu V(x)u=lambda f(x)u+g(x)|u|^{p-2}uquad text{ in }mathbb{R}^{N}, end{equation*}% where $Ngeq 3,2 0$ and $mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0 0$ is the principal eigenvalue of $-Delta $ in $H_{0}^{1}(Omega )$ with weight function $f_{Omega }:=f|_{Omega }$, and $phi _{1}>0$ is the corresponding principal eigenfunction. When $Ngeq 3$ and $2 0$ small and $0 0$ small and $lambda _{1}(f_{Omega })leq lambda 0$, at least two positive solutions exist for $a>a_{0}(p)$ and $lambda^{+}_{a} 0$ and $lambda^{+}_{a}geq0$.
我们研究一类涉及线性项和超线性项组合的Kirchhoff型方程，如下所示: begin{equation*} -left( aint_{mathbb{R}^{N}}|nabla u|^{2}dx+1right) Delta u+mu V(x)u=lambda f(x)u+g(x)|u|^{p-2}uquad text{ in }mathbb{R}^{N}, end{equation*}% where $Ngeq 3,2 0$ and $mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0 0$ is the principal eigenvalue of $-Delta $ in $H_{0}^{1}(Omega )$ with weight function $f_{Omega }:=f|_{Omega }$, and $phi _{1}>0$ is the corresponding principal eigenfunction. When $Ngeq 3$ and $2 0$ small and $0 0$ small and $lambda _{1}(f_{Omega })leq lambda 0$, at least two positive solutions exist for $a>a_{0}(p)$ and $lambda^{+}_{a} 0$ and $lambda^{+}_{a}geq0$.
{"title":"On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms","authors":"Juntao Sun, Kuan‐Hsiang Wang, Tsung‐fang Wu","doi":"10.1063/5.0030427","DOIUrl":"https://doi.org/10.1063/5.0030427","url":null,"abstract":"We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: begin{equation*} -left( aint_{mathbb{R}^{N}}|nabla u|^{2}dx+1right) Delta u+mu V(x)u=lambda f(x)u+g(x)|u|^{p-2}uquad text{ in }mathbb{R}^{N}, end{equation*}% where $Ngeq 3,2 0$ and $mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0 0$ is the principal eigenvalue of $-Delta $ in $H_{0}^{1}(Omega )$ with weight function $f_{Omega }:=f|_{Omega }$, and $phi _{1}>0$ is the corresponding principal eigenfunction. When $Ngeq 3$ and $2 0$ small and $0 0$ small and $lambda _{1}(f_{Omega })leq lambda 0$, at least two positive solutions exist for $a>a_{0}(p)$ and $lambda^{+}_{a} 0$ and $lambda^{+}_{a}geq0$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83084671","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 consider the fourth order Schr"odinger operator $H=Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ then the wave operators $W_{pm}= s,-,lim_{tto pm infty} e^{itH}e^{-itH_0}$ extend to bounded operators on $L^p(mathbb R^3)$ for all $1
{"title":"On the $L^p$ boundedness of the wave operators for fourth order Schrödinger operators","authors":"M. Goldberg, William R. Green","doi":"10.1090/tran/8377","DOIUrl":"https://doi.org/10.1090/tran/8377","url":null,"abstract":"We consider the fourth order Schr\"odinger operator $H=Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ then the wave operators $W_{pm}= s,-,lim_{tto pm infty} e^{itH}e^{-itH_0}$ extend to bounded operators on $L^p(mathbb R^3)$ for all $1","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86884786","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 : 2020-08-16DOI: 10.1016/J.MATPUR.2021.05.005
Mihalis Mourgoglou, Carmelo Puliatti
{"title":"Blow-ups of caloric measure in time varying domains and applications to two-phase problems","authors":"Mihalis Mourgoglou, Carmelo Puliatti","doi":"10.1016/J.MATPUR.2021.05.005","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.05.005","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84072822","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 : 2020-08-15DOI: 10.1016/J.MATPUR.2021.07.003
Jinping Zhuge
{"title":"Regularity of a transmission problem and periodic homogenization","authors":"Jinping Zhuge","doi":"10.1016/J.MATPUR.2021.07.003","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.07.003","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77692673","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 : 2020-08-15DOI: 10.14232/EJQTDE.2020.1.79
E. Crooks, M. Grinfeld
We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.
研究标量单稳定反应扩散方程中行波解的显式存在性与最小速度行波之间的联系。
{"title":"Minimal travelling wave speed and explicit solutions in monostable reaction-diffusion equations","authors":"E. Crooks, M. Grinfeld","doi":"10.14232/EJQTDE.2020.1.79","DOIUrl":"https://doi.org/10.14232/EJQTDE.2020.1.79","url":null,"abstract":"We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88923855","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}
For a parameter $gammain(1,2)$, we study the fully nonlinear version of the Alt-Phillips equation, $F(D^2u)=u^{gamma-1}$, for $uge 0.$ We establish the optimal regularity of the solution, as well as the $C^1$ regularity of the regular part of the free boundary.
{"title":"On the Fully Nonlinear Alt–Phillips Equation","authors":"Yijing Wu, Hui Yu","doi":"10.1093/IMRN/RNAA359","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA359","url":null,"abstract":"For a parameter $gammain(1,2)$, we study the fully nonlinear version of the Alt-Phillips equation, $F(D^2u)=u^{gamma-1}$, for $uge 0.$ We establish the optimal regularity of the solution, as well as the $C^1$ regularity of the regular part of the free boundary.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81877246","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 : 2020-08-14DOI: 10.1016/J.MATPUR.2021.05.002
D. Gérard-Varet
{"title":"Derivation of the Batchelor-Green formula for random suspensions","authors":"D. Gérard-Varet","doi":"10.1016/J.MATPUR.2021.05.002","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.05.002","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"19 9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83215192","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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