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Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation 广义无粘面拟地转方程的移动解和旋转解
arXiv: Analysis of PDEs Pub Date : 2020-08-29 DOI: 10.1090/TRAN/8406
Weiwei Ao, J. Dávila, Manuel del Pino, M. Musso, Juncheng Wei
{"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":null,"pages":null},"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}
引用次数: 26
Large time behavior and diffusion limit for a system of balance laws from chemotaxis in multi-dimensions 多维趋化性平衡律系统的大时间行为和扩散极限
arXiv: Analysis of PDEs Pub Date : 2020-08-25 DOI: 10.4310/CMS.2021.V19.N1.A10
Tong Li, Dehua Wang, Fang Wang, Zhian Wang, Kun Zhao
{"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":null,"pages":null},"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}
引用次数: 3
Generic Dynamical Properties of Connections on Vector Bundles 向量束上连接的一般动力学性质
arXiv: Analysis of PDEs Pub Date : 2020-08-20 DOI: 10.1093/IMRN/RNAB069
Mihajlo Ceki'c, Thibault Lefeuvre
{"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":null,"pages":null},"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}
引用次数: 7
On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms 线性项和超线性项联合作用下的不定kirchhoff型方程
arXiv: Analysis of PDEs Pub Date : 2020-08-19 DOI: 10.1063/5.0030427
Juntao Sun, Kuan‐Hsiang Wang, Tsung‐fang Wu
{"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":null,"pages":null},"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}
引用次数: 3
On the $L^p$ boundedness of the wave operators for fourth order Schrödinger operators 四阶Schrödinger算子的波算子的L^p有界性
arXiv: Analysis of PDEs Pub Date : 2020-08-17 DOI: 10.1090/tran/8377
M. Goldberg, William R. Green
{"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":null,"pages":null},"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}
引用次数: 7
Blow-ups of caloric measure in time varying domains and applications to two-phase problems 时变域热测量的爆破及其在两相问题中的应用
arXiv: Analysis of PDEs Pub Date : 2020-08-16 DOI: 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":null,"pages":null},"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}
引用次数: 4
Regularity of a transmission problem and periodic homogenization 传输问题的正则性与周期均匀化
arXiv: Analysis of PDEs Pub Date : 2020-08-15 DOI: 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":null,"pages":null},"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}
引用次数: 7
Minimal travelling wave speed and explicit solutions in monostable reaction-diffusion equations 单稳定反应扩散方程的最小行波速度和显式解
arXiv: Analysis of PDEs Pub Date : 2020-08-15 DOI: 10.14232/EJQTDE.2020.1.79
E. Crooks, M. Grinfeld
{"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":null,"pages":null},"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}
引用次数: 3
On the Fully Nonlinear Alt–Phillips Equation 关于全非线性Alt-Phillips方程
arXiv: Analysis of PDEs Pub Date : 2020-08-14 DOI: 10.1093/IMRN/RNAA359
Yijing Wu, Hui Yu
{"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":null,"pages":null},"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}
引用次数: 3
Derivation of the Batchelor-Green formula for random suspensions 随机悬架的Batchelor-Green公式的推导
arXiv: Analysis of PDEs Pub Date : 2020-08-14 DOI: 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":null,"pages":null},"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}
引用次数: 13
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