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On blowup for the supercritical quadratic wave equation 超临界二次波方程的爆破问题
arXiv: Analysis of PDEs Pub Date : 2021-09-24 DOI: 10.5445/IR/1000138775
E. Csobo, Irfan Glogi'c, Birgit Schorkhuber
{"title":"On blowup for the supercritical quadratic wave equation","authors":"E. Csobo, Irfan Glogi'c, Birgit Schorkhuber","doi":"10.5445/IR/1000138775","DOIUrl":"https://doi.org/10.5445/IR/1000138775","url":null,"abstract":"We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d ge 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^∗$ which exists for all d $d ge 7$. For $d = 9$, we study the stability of $u^∗$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^*$ . In similarity coordinates, this family represents a co-dimension one Lipschitz manifold modulo translation symmetries. In addition, in $d = 7$ and $d = 9$, we prove non-radial stability of the well-known ODE blowup solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80145171","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
Global wellposedness of NLS in $H^1(mathbb{R})+H^s(mathbb{T})$ $H^1(mathbb{R})+H^s(mathbb{T})$中NLS的全局适定性
arXiv: Analysis of PDEs Pub Date : 2021-09-23 DOI: 10.5445/IR/1000137946
Friedrich Klaus, P. Kunstmann
{"title":"Global wellposedness of NLS in $H^1(mathbb{R})+H^s(mathbb{T})$","authors":"Friedrich Klaus, P. Kunstmann","doi":"10.5445/IR/1000137946","DOIUrl":"https://doi.org/10.5445/IR/1000137946","url":null,"abstract":"We show global wellposedness for the defocusing cubic nonlinear Schrodinger equation (NLS) in $H^1(mathbb{R}) + H^{3/2+}(mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(mathbb{R}) + H^{5/2+}(mathbb{T})$. This complements local results for the cubic NLS [6] and global results for the quadratic NLS \u0000[8] in this hybrid setting.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84939408","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
Well-posedness for Maxwell equations with Kerr nonlinearity in three dimensions via Strichartz estimates 基于Strichartz估计的三维克尔非线性Maxwell方程的适定性
arXiv: Analysis of PDEs Pub Date : 2021-08-17 DOI: 10.5445/IR/1000136611
R. Schippa
{"title":"Well-posedness for Maxwell equations with Kerr nonlinearity in three dimensions via Strichartz estimates","authors":"R. Schippa","doi":"10.5445/IR/1000136611","DOIUrl":"https://doi.org/10.5445/IR/1000136611","url":null,"abstract":"We show new local well-posedness results for quasilinear Maxwell equations in three spatial dimensions with an emphasis on the Kerr nonlinearity. For this purpose, new Strichartz estimates are proved for solutions with rough permittivity by conjugation to half-wave equations. We use the Strichartz estimates in a known combination with energy estimates to derive the new well-posedness results.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73876678","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: 1
Higher-order synchronization of a nudging-based algorithm for data assimilation for the 2D NSE: a refined paradigm for global interpolant observables 用于二维NSE数据同化的基于轻推算法的高阶同步:用于全局插值可观测值的改进范例
arXiv: Analysis of PDEs Pub Date : 2021-08-11 DOI: 10.13016/M2FYO1-BNGF
A. Biswas, K. Brown, V. Martinez
{"title":"Higher-order synchronization of a nudging-based algorithm for data assimilation for the 2D NSE: a refined paradigm for global interpolant observables","authors":"A. Biswas, K. Brown, V. Martinez","doi":"10.13016/M2FYO1-BNGF","DOIUrl":"https://doi.org/10.13016/M2FYO1-BNGF","url":null,"abstract":"This paper considers a nudging-based scheme for data assimilation for the two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the synchronization of the signal produced by this algorithm with the true signal, to which the observations correspond, in all higher-order Sobolev topologies. This work complements previous results in the literature where conditions were identified under which synchronization is guaranteed either with respect to only the $H^1$--topology, in the case of general observables, or to the analytic Gevrey topology, in the case of spectral observables. To accommodate the property of synchronization in the stronger topologies, the framework of general interpolant observable operators, originally introduced by Azouani, Olson, and Titi, is expanded to a far richer class of operators. A significant effort is dedicated to the development of this more expanded framework, specifically, their basic approximation properties, the identification of subclasses of such operators relevant to obtaining synchronization, as well as the detailed relation between the structure of these operators and the system regarding the syncrhonization property. One of the main features of this framework is its \"mesh-free\" aspect, which allows the observational data itself to dictate the subdivision of the domain. Lastly, estimates for the radius of the absorbing ball of the 2D NSE in all higher-order Sobolev norms are obtained, thus properly generalizing previously known bounds; such estimates are required for establishing the synchronization property of the algorithm in the higher-order topologies.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79366587","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: 1
Fourier transform of surface-carried measures of two-dimensional generic surfaces and applications 二维一般曲面的面载测度的傅里叶变换及其应用
arXiv: Analysis of PDEs Pub Date : 2021-07-28 DOI: 10.5445/IR/1000136612
Jean-Claude Cuenin, R. Schippa
{"title":"Fourier transform of surface-carried measures of two-dimensional generic surfaces and applications","authors":"Jean-Claude Cuenin, R. Schippa","doi":"10.5445/IR/1000136612","DOIUrl":"https://doi.org/10.5445/IR/1000136612","url":null,"abstract":"We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schrodinger operators on the cubic lattice.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88487449","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
Biharmonic nonlinear scalar field equations 双调和非线性标量场方程
arXiv: Analysis of PDEs Pub Date : 2021-07-15 DOI: 10.5445/IR/1000135513
Jarosław Mederski, Jakub Siemianowski
{"title":"Biharmonic nonlinear scalar field equations","authors":"Jarosław Mederski, Jakub Siemianowski","doi":"10.5445/IR/1000135513","DOIUrl":"https://doi.org/10.5445/IR/1000135513","url":null,"abstract":"We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinearequation $$Delta^2u=g(x,u)qquadtext{ in }mathbb{R}^N$$ with a Caratheodory function $g:mathbb{R}^Ntimesmathbb{R}tomathbb{R}$, $Nge5$. The regularity results give rise to the existence of ground state solutions provided that g has a general subcritical growth at infinity. We also conceive a newbiharmonic logarithmic Sobolev inequality$$int_{mathbb{R}^N}|u|^2log|u|,dx le frac{N}{8}logleft(Cint_{mathbb{R}^N}|Delta u|^2,dxright), quadtext{ for } uin H^2(mathbb{R}^N), int_{mathbb{R}^N}u^2,dx=1,$$for a constant $0<C<left(frac{2}{pi e N}right)^2$ and we characterize its minimizers.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73285665","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 Viscous Cahn-Hilliard-Oono System with Chemotaxis and Singular Potential 具有趋化性和奇异势的粘性Cahn-Hilliard-Oono系统
arXiv: Analysis of PDEs Pub Date : 2021-05-29 DOI: 10.22541/AU.162227013.31240680/V1
Jingning He
{"title":"On the Viscous Cahn-Hilliard-Oono System with Chemotaxis and Singular Potential","authors":"Jingning He","doi":"10.22541/AU.162227013.31240680/V1","DOIUrl":"https://doi.org/10.22541/AU.162227013.31240680/V1","url":null,"abstract":"We analyze a diffuse interface model that couples a viscous\u0000Cahn-Hilliard equation for the phase variable with a diffusion-reaction\u0000equation for the nutrient concentration. The system under consideration\u0000also takes into account some important mechanisms like chemotaxis,\u0000active transport as well as nonlocal interaction of Oono’s type. When\u0000the spatial dimension is three, we prove the existence and uniqueness of\u0000global weak solutions to the model with singular potentials including\u0000the physically relevant logarithmic potential. Then we obtain some\u0000regularity properties of the weak solutions when t>0. In\u0000particular, with the aid of the viscous term, we prove the so-called\u0000instantaneous separation property of the phase variable such that it\u0000stays away from the pure states ±1 as long as t>0.\u0000Furthermore, we study long-time behavior of the system, by proving the\u0000existence of a global attractor and characterizing its ω-limit set.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89963886","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: 1
On smoothing estimates in modulation spaces and the NLS with slowly decaying initial data 调制空间中的平滑估计和初始数据缓慢衰减的NLS
arXiv: Analysis of PDEs Pub Date : 2021-05-05 DOI: 10.5445/IR/1000132421
R. Schippa
{"title":"On smoothing estimates in modulation spaces and the NLS with slowly decaying initial data","authors":"R. Schippa","doi":"10.5445/IR/1000132421","DOIUrl":"https://doi.org/10.5445/IR/1000132421","url":null,"abstract":"We show new local $L^p$-smoothing estimates for the Schrodinger equation with initial data in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and $L^p$-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in $L^p(mathbb{R}^d)$ for $d ge 1$ and $p>2$, which was previously known for $d ge 2$. The estimates are applied to show new local and global well-posedness results for the cubic nonlinear Schrodinger equation on the line. Lastly, we show $ell^2$ -decoupling inequalities for variable-coefficient versions of elliptic and non-elliptic Schrodinger phase functions.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88179520","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
Breather solutions for a quasilinear (1+1)-dimensional wave equation 拟线性(1+1)维波动方程的呼吸解
arXiv: Analysis of PDEs Pub Date : 2021-04-26 DOI: 10.5445/IR/1000132263
Simon Kohler, Wolfgang Reichel Institute for Analysis, Karlsruhe Institute of Technology, D. Karlsruhe, H Germany
{"title":"Breather solutions for a quasilinear (1+1)-dimensional wave equation","authors":"Simon Kohler, Wolfgang Reichel Institute for Analysis, Karlsruhe Institute of Technology, D. Karlsruhe, H Germany","doi":"10.5445/IR/1000132263","DOIUrl":"https://doi.org/10.5445/IR/1000132263","url":null,"abstract":"We consider the $(1 + 1)$-dimensional quasilinear wave equation $g(x)w_{tt} − w_{xx} + h(x)(w^3_t)_t = 0$ on $mathbb{R}timesmathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $gin L^{infty}(mathbb{R})$ is even with $gnotequiv 0$ and $h(x) = gammadelta_0(x)$ with $gammainmathbb{R}backslash{0}$ and $delta_0$ the delta distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k = frac{d^2}{dx^2}-k^2omega^2g$ on $L^2(mathbb{R})$ for all $kin 2mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitely given step potentials and periodic step potentials $g$. In these examples we even find infinitely many distinct breathers.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78098018","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: 1
Global results for a Cauchy problem related to biharmonic wave maps 关于双调和波映射的Cauchy问题的全局结果
arXiv: Analysis of PDEs Pub Date : 2021-02-25 DOI: 10.5445/IR/1000130150
Tobias Schmid
{"title":"Global results for a Cauchy problem related to biharmonic wave maps","authors":"Tobias Schmid","doi":"10.5445/IR/1000130150","DOIUrl":"https://doi.org/10.5445/IR/1000130150","url":null,"abstract":"We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space $$dot{B}^{2,1}_{frac{d}{2}}(mathbb{R}^d) times dot{B}^{2,1}_{frac{d}{2}-2}(mathbb{R}^d)$$ for $ d geq 3 $. Since the solution persists higher regularity of the initial data, we obtain a small data global regularity result for the biharmonic wave maps equation for a certain class of target manifolds including the sphere.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83261233","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: 1
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