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Approximation of divergence-free vector fields vanishing on rough planar sets 粗糙平面集上消失的无发散向量场的近似值
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-15 DOI: arxiv-2409.09880
Giacomo Del Nin, Bian Wu
{"title":"Approximation of divergence-free vector fields vanishing on rough planar sets","authors":"Giacomo Del Nin, Bian Wu","doi":"arxiv-2409.09880","DOIUrl":"https://doi.org/arxiv-2409.09880","url":null,"abstract":"Given any divergence-free vector field of Sobolev class $W^{m,p}_0(Omega)$\u0000in bounded open subset $Omega subset mathbb{R}^2$, we are interested in\u0000approximating it in $W^{m,p}$ with divergence-free smooth vector fields\u0000compactly supported in $Omega$. We show that this approximation property holds\u0000in the following cases. For $p>2$, this holds given that $partial Omega$ has\u0000zero Lebesgue measure (a weaker but more technical condition is sufficient);\u0000For $p leq 2$, this holds if $Omega^c$ can be decomposed into finitely many\u0000disjoint closed set, each of which is connected or $d$-Ahlfors regular for some\u0000$din[0,2]$. This has links to the uniqueness of weak solutions to the Stokes\u0000equation in $Omega$. For H\"older spaces, we prove this property in general\u0000bounded domains.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260684","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: 0
Upper bounds of nodal sets for Gevrey regular parabolic equations Gevrey 正则抛物方程的节点集上界
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-15 DOI: arxiv-2409.09879
Guher Camliyurt, Igor Kukavica, Linfeng Li
{"title":"Upper bounds of nodal sets for Gevrey regular parabolic equations","authors":"Guher Camliyurt, Igor Kukavica, Linfeng Li","doi":"arxiv-2409.09879","DOIUrl":"https://doi.org/arxiv-2409.09879","url":null,"abstract":"We consider the size of the nodal set of the solution of the second order\u0000parabolic-type equation with Gevrey regular coefficients. We provide an upper\u0000bound as a function of time. The dependence agrees with a sharp upper bound\u0000when the coefficients are analytic.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260683","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: 0
Small scales in inviscid limits of steady fluids 稳定流体不粘性极限中的小尺度
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-15 DOI: arxiv-2409.09604
Yan Guo, Zhuolun Yang
{"title":"Small scales in inviscid limits of steady fluids","authors":"Yan Guo, Zhuolun Yang","doi":"arxiv-2409.09604","DOIUrl":"https://doi.org/arxiv-2409.09604","url":null,"abstract":"In this article, we study the 2D incompressible steady Navier-Stokes equation\u0000in a channel $(-L,0)times(-1,1)$ with the no-slip boundary condition on ${Y =\u0000pm 1}$, and consider the inviscid limit $varepsilon to 0$. In the special\u0000case of Euler shear flow $(u_e(Y),0)$, we construct a steady Navier-Stokes\u0000solution for $varepsilon ll1$, $$left{ begin{aligned} &u^varepsilon sim\u0000u_e + u_p + O(sqrt{varepsilon}), &v^varepsilon sim h(Y)\u0000exp{Xu_e(Y)/varepsilon} + O(sqrt{varepsilon}), end{aligned}right. $$\u0000where $u_p$ represents the classical Prandtl layer profile, and $h(Y)$ is an\u0000arbitrary smooth, compactly-supported function with small magnitude. While the\u0000classical Prandtl boundary layer $u_p$ exhibits a small scale of order\u0000$sqrt{varepsilon}$ in $Y$ near $Y = pm 1$, the profile we construct reveals\u0000an $varepsilon$ small scale of $Xu_e(Y)$ in the vertical velocity component.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260703","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: 0
On scattering for two-dimensional quintic Schrödinger equation under partial harmonic confinement 部分谐波约束下二维五元薛定谔方程的散射问题
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-15 DOI: arxiv-2409.09789
Zuyu Ma, Yilin Song, Ruixiao Zhang, Zehua Zhao, Jiqiang Zheng
{"title":"On scattering for two-dimensional quintic Schrödinger equation under partial harmonic confinement","authors":"Zuyu Ma, Yilin Song, Ruixiao Zhang, Zehua Zhao, Jiqiang Zheng","doi":"arxiv-2409.09789","DOIUrl":"https://doi.org/arxiv-2409.09789","url":null,"abstract":"In this article, we study the scattering theory for the two dimensional\u0000defocusing quintic nonlinear Schr\"odinger equation(NLS) with partial harmonic\u0000oscillator which is given by begin{align}label{NLS-abstract}\u0000begin{cases}tag{PHNLS}\u0000ipartial_tu+(partial_{x_1}^2+partial_{x_2}^2)u-x_2^2u=|u|^4u,&(t,x_1,x_2)inmathbb{R}timesmathbb{R}timesmathbb{R},\u0000u(0,x_1,x_2)=u_0(x_1,x_2). end{cases} end{align} First, we establish the linear profile decomposition for the Schr\"odinger\u0000operator $e^{it(partial_{x_1}^2+partial_{x_2}^2-x_2^2)}$ by utilizing the\u0000classical linear profile decomposition associated with the Schr\"odinger\u0000equation in $L^2(mathbb{R})$. Then, applying the normal form technique, we\u0000approximate the nonlinear profiles using solutions of the new-type quintic\u0000dispersive continuous resonant (DCR) system. This allows us to employ the\u0000concentration-compactness/rigidity argument introduced by Kenig and Merle in\u0000our setting and prove scattering for equation (PHNLS) in the weighted Sobolev\u0000space. The second part of this paper is dedicated to proving the scattering theory\u0000for this mass-critical (DCR) system. Inspired by Dodson's seminal work [B.\u0000Dodson, Amer. J. Math. 138 (2016), 531-569], we develop long-time Strichartz\u0000estimates associated with the spectral projection operator $Pi_n$, along with\u0000low-frequency localized Morawetz estimates, to address the challenges posed by\u0000the Galilean transformation and spatial translation.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260700","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: 0
On equilibrium in control problems with applications to evolution systems 论控制问题中的均衡，并应用于演化系统
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-15 DOI: arxiv-2409.09805
Radu Precup, Andrei Stan
{"title":"On equilibrium in control problems with applications to evolution systems","authors":"Radu Precup, Andrei Stan","doi":"arxiv-2409.09805","DOIUrl":"https://doi.org/arxiv-2409.09805","url":null,"abstract":"In this paper we examine a mutual control problem for systems of two abstract\u0000evolution equations subject to a proportionality final condition. Related\u0000observability and semi-observability problems are discussed. The analysis\u0000employs a vector fixed-point approach, using matrices rather than constants,\u0000and applies the technique of Bielecki equivalent norms.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260682","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: 0
A new blowup criterion for strong solutions of a coupled periodic Camassa-Holm system 卡马萨-霍姆耦合周期系统强解的新炸毁准则
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-15 DOI: arxiv-2409.09762
Yonghui Zhou, Xiaowan Li
{"title":"A new blowup criterion for strong solutions of a coupled periodic Camassa-Holm system","authors":"Yonghui Zhou, Xiaowan Li","doi":"arxiv-2409.09762","DOIUrl":"https://doi.org/arxiv-2409.09762","url":null,"abstract":"This paper is concerned with the wave breaking phenomena for a coupled\u0000periodic Camassa-Holm system. We establish a new blowup criterion for strong\u0000solutions by the method of characteristic and convolution estimates, and also\u0000give the existence interval of the blowup point.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"207 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260701","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: 0
Existence of solutions for some systems of superdiffusive integro-differential equations in population dynamics depending on the natality and mortality rates 取决于出生率和死亡率的某些人口动力学超扩散积分微分方程系统解的存在性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-14 DOI: arxiv-2409.09507
Vitali Vougalter
{"title":"Existence of solutions for some systems of superdiffusive integro-differential equations in population dynamics depending on the natality and mortality rates","authors":"Vitali Vougalter","doi":"arxiv-2409.09507","DOIUrl":"https://doi.org/arxiv-2409.09507","url":null,"abstract":"We prove the existence of stationary solutions for some systems of\u0000reaction-diffusion type equations with superdiffusion in the corresponding H^2\u0000spaces. Our method is based on the fixed point theorem when the elliptic\u0000problems contain first order differential operators with and without the\u0000Fredholm property, which may depend on the outcome of the competition between\u0000the natality and the mortality rates contained in the equations of our systems.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260702","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: 0
A shape optimization problem in cylinders and related overdetermined problems 圆柱体形状优化问题及相关超定问题
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-14 DOI: arxiv-2409.09448
Paolo Caldiroli, Alessandro Iacopetti, Filomena Pacella
{"title":"A shape optimization problem in cylinders and related overdetermined problems","authors":"Paolo Caldiroli, Alessandro Iacopetti, Filomena Pacella","doi":"arxiv-2409.09448","DOIUrl":"https://doi.org/arxiv-2409.09448","url":null,"abstract":"In this paper, we study a shape optimization problem for the torsional energy\u0000associated with a domain contained in an infinite cylinder, under a volume\u0000constraint. We prove that a minimizer exists for all fixed volumes and show\u0000some of its geometric and topological properties. As this issue is closely\u0000related to the question of characterizing domains in cylinders that admit\u0000solutions to an overdetermined problem, our minimization result allows us to\u0000deduce interesting consequences in that direction. In particular, we find that,\u0000for some cylinders and some volumes, the ``trivial\" domain given by a bounded\u0000cylinder is not the only domain where the overdetermined problem has a\u0000solution. Moreover, it is not even a minimizer, which indicates that solutions\u0000with flat level sets are not always the best candidates for optimizing the\u0000torsional energy.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260707","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: 0
A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations 欧拉方程弱解的流体动力压力双霍尔德正则性说明
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-14 DOI: arxiv-2409.09433
Siran Li, Ya-Guang Wang
{"title":"A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations","authors":"Siran Li, Ya-Guang Wang","doi":"arxiv-2409.09433","DOIUrl":"https://doi.org/arxiv-2409.09433","url":null,"abstract":"We give an elementary proof for the double H\"{o}lder regularity of the\u0000hydrodynamic pressure for weak solutions of the Euler Equations in a bounded\u0000$C^2$-domain $Omega subset mathbb{R}^d$; $dgeq 3$. That is, for velocity $u\u0000in C^{0,gamma}(Omega;mathbb{R}^d)$ with some $0<gamma<1/2$, we show that\u0000the pressure $p in C^{0,2gamma}(Omega)$. This is motivated by the studies of\u0000turbulence and anolalous dissipation in mathematical hydrodynamics and,\u0000recently, has been established in [L. De Rosa, M. Latocca, and G. Stefani, Int.\u0000Math. Res. Not. 2024.3 (2024), 2511-2560] over $C^{2,alpha}$-domains by means\u0000of pseudodifferential calculus. Our approach involves only standard elliptic\u0000PDE techniques, and relies crucially on the modified pressure introduced in [C.\u0000W. Bardos, D. W. Boutros, and E. S. Titi, H\"{o}lder regularity of the pressure\u0000for weak solutions of the 3D Euler equations in bounded domains, arXiv:\u00002304.01952] and the potential estimates in [L. Silvestre, unpublished notes].\u0000The key novel ingredient of our proof is the introduction of two cutoff\u0000functions whose localisation parameters are carefully chosen as a power of the\u0000distance to $partialOmega$.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260705","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: 0
Local-in-time analytic solutions for an inviscid model of superfluidity in 3D 三维超流动无粘性模型的局部时间解析解
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-14 DOI: arxiv-2409.09404
Pranava Chaitanya Jayanti
{"title":"Local-in-time analytic solutions for an inviscid model of superfluidity in 3D","authors":"Pranava Chaitanya Jayanti","doi":"arxiv-2409.09404","DOIUrl":"https://doi.org/arxiv-2409.09404","url":null,"abstract":"We address the existence of solutions for the inviscid version of the\u0000Hall-Vinen-Bekharevich-Khalatnikov equations in 3D, a macro-scale model of\u0000superfluidity. This system couples the incompressible Euler equations for the\u0000normal fluid and superfluid using a nonlinear mutual friction term that acts\u0000only at points of non-zero superfluid vorticity. In the first rigorous study of\u0000the inviscid HVBK system, we construct a unique local-in-time solution that is\u0000analytic in time and space.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260706","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: 0
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