Pub Date : 2024-02-14DOI: 10.1007/s10473-024-0317-6
Han Wang, Yinghui Zhang
We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular, are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal L2 convergence rate of the k (∈ [0, 3])-order spatial derivatives of the solution is ({(1 + t)^{ - {{3 + 2k} over 4}}}). Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.
{"title":"The optimal large time behavior of 3D quasilinear hyperbolic equations with nonlinear damping","authors":"Han Wang, Yinghui Zhang","doi":"10.1007/s10473-024-0317-6","DOIUrl":"https://doi.org/10.1007/s10473-024-0317-6","url":null,"abstract":"<p>We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular, are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal <i>L</i><sup>2</sup> convergence rate of the <i>k</i> (∈ [0, 3])-order spatial derivatives of the solution is <span>({(1 + t)^{ - {{3 + 2k} over 4}}})</span>. Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"17 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-14DOI: 10.1007/s10473-024-0303-z
Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng
Let X and Y be two normed spaces. Let ({cal U}) be a non-principal ultrafilter on ℕ. Let g: X → Y be a standard ε-phase isometry for some ε ≥ 0, i.e., g(0) = 0, and for all u, v ϵ X,
The mapping g is said to be a phase isometry provided that ε = 0. In this paper, we show the following universal inequality of g: for each ({u^ * } in {w^ * } - exp ,,||{u^ * }||{B_{{X^ * }}}), there exist a phase function ({sigma _{{u^ * }}}:X to { - 1,1} ) and φ ϵ Y* with (||varphi || = ||{u^ * }|| equiv alpha ) satisfying that
$$|leftlangle {{u^ * },u} rightrangle - {sigma _{{u^ * }}}(u)leftlangle {varphi ,g(u)} rightrangle | le {5 over 2}varepsilon alpha ,,,,{rm{for}},{rm{all}},u in X.$$
In particular, let X be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all u* ∈ X*; (2) the constant ({5 over 2}) can be reduced to ({3 over 2}) provided that Y* is strictly convex; (3) the existence of such a g implies the existence of a phase isometry Θ: X → Y such that (Theta (u) = mathop {lim }limits_{n,{cal U}} {{g(nu)} over n}) provided that Y** has the w*-Kadec-Klee property (for example, Y is both reflexive and locally uniformly convex).
让 X 和 Y 是两个规范空间。让 ({cal U}) 是ℕ上的一个非主超滤波器。对于某个 ε ≥ 0,让 g: X → Y 是一个标准的 ε 相等距,即g(0) = 0,并且对于所有 u、v ϵ X,$$||,,|,||g(u) + g(v)|| pm |g(u) - g(v)||,| - |,|u + v|| pm ||u - v||,|,,|,le varepsilon .$$只要 ε = 0,映射 g 就被称为相等几何。在本文中,我们证明了 g 的以下普遍不等式:对于每个 ({u^ * } in {w^ * } - exp ,,||{u^ * }||{B_{{X^ * }}}), 都存在一个相位函数 ({sigma _{u^ * }}}:X to { - 1,1} ) and φ ϵ Y* with (||varphi || = ||{u^ * }|| equiv alpha ) satisfying that $$|leftlangle {{u^ * }、u} - {sigma _{{u^ * }}}(u)leftlangle {varphi ,g(u)} rightrangle | le {5 over 2}varepsilon alpha ,,,{rm{for},{rm{all}},u in X.$$特别地,让 X 是一个光滑的巴拿赫空间。然后我们证明以下几点:(1) 对于所有u*∈X*,普遍不等式都成立;(2) 常量({5 over 2})可以简化为({3 over 2}),前提是Y*是严格凸的;(3) 这样一个g的存在意味着相等几何Θ的存在:X → Y such that (θ (u) = mathop {lim }limits_{n,{cal U}}{{g(nu)}overn}/),条件是 Y** 具有 w*-Kadec-Klee 属性(例如,Y 既是反折的,又是局部均匀凸的)。
{"title":"On a universal inequality for approximate phase isometries","authors":"Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng","doi":"10.1007/s10473-024-0303-z","DOIUrl":"https://doi.org/10.1007/s10473-024-0303-z","url":null,"abstract":"<p>Let <i>X</i> and <i>Y</i> be two normed spaces. Let <span>({cal U})</span> be a non-principal ultrafilter on ℕ. Let g: <i>X</i> → <i>Y</i> be a standard <i>ε</i>-phase isometry for some <i>ε</i> ≥ 0, i.e., <i>g</i>(0) = 0, and for all <i>u, v ϵ X</i>, </p><span>$$|,,|,||g(u) + g(v)|| pm ||g(u) - g(v)||,| - |,||u + v|| pm ||u - v||,|,,|, le varepsilon .$$</span><p>The mapping <i>g</i> is said to be a phase isometry provided that <i>ε</i> = 0. In this paper, we show the following universal inequality of <i>g</i>: for each <span>({u^ * } in {w^ * } - exp ,,||{u^ * }||{B_{{X^ * }}})</span>, there exist a phase function <span>({sigma _{{u^ * }}}:X to { - 1,1} )</span> and <i>φ</i> ϵ <i>Y</i>* with <span>(||varphi || = ||{u^ * }|| equiv alpha )</span> satisfying that </p><span>$$|leftlangle {{u^ * },u} rightrangle - {sigma _{{u^ * }}}(u)leftlangle {varphi ,g(u)} rightrangle | le {5 over 2}varepsilon alpha ,,,,{rm{for}},{rm{all}},u in X.$$</span><p>In particular, let <i>X</i> be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all <i>u</i>* ∈ <i>X</i>*; (2) the constant <span>({5 over 2})</span> can be reduced to <span>({3 over 2})</span> provided that <i>Y</i>* is strictly convex; (3) the existence of such a g implies the existence of a phase isometry Θ: <i>X</i> → <i>Y</i> such that <span>(Theta (u) = mathop {lim }limits_{n,{cal U}} {{g(nu)} over n})</span> provided that <i>Y</i>** has the <i>w</i>*-Kadec-Klee property (for example, <i>Y</i> is both reflexive and locally uniformly convex).</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-14DOI: 10.1007/s10473-024-0311-z
Cuntao Xiao, Hua Qiu, Zheng-an Yao
In this paper, we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms. We establish the global existence of mild solutions to this system with small initial data. In addition, we also obtain the Gevrey class regularity and the temporal decay rate of the solution.
{"title":"The global existence and analyticity of a mild solution to the 3D regularized MHD equations","authors":"Cuntao Xiao, Hua Qiu, Zheng-an Yao","doi":"10.1007/s10473-024-0311-z","DOIUrl":"https://doi.org/10.1007/s10473-024-0311-z","url":null,"abstract":"<p>In this paper, we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms. We establish the global existence of mild solutions to this system with small initial data. In addition, we also obtain the Gevrey class regularity and the temporal decay rate of the solution.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}