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Ill-Posedness for the Cauchy Problem of the Modified Camassa-Holm Equation in (B_{infty ,1}^0) 修正卡马萨-霍姆方程在(B_{infty ,1}^0) 中的考奇问题的非确定性
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-10-19 DOI: 10.1007/s00021-024-00903-1
Zhen He, Zhaoyang Yin
{"title":"Ill-Posedness for the Cauchy Problem of the Modified Camassa-Holm Equation in (B_{infty ,1}^0)","authors":"Zhen He,&nbsp;Zhaoyang Yin","doi":"10.1007/s00021-024-00903-1","DOIUrl":"10.1007/s00021-024-00903-1","url":null,"abstract":"<div><p>In this paper, we prove the norm inflation and get the ill-posedness for the modified Camassa-Holm equation in <span>(B_{infty ,1}^0)</span>. Therefore we completed all well-posedness and ill-posedness problem for the modified Camassa-Holm equation in all critical spaces <span>(B_{p,1}^frac{1}{p})</span> with <span>(pin [1,infty ])</span>.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142451083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Time Evolution of the Navier–Stokes Flow in Far-Field 远场纳维-斯托克斯流的时间演变
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-10-16 DOI: 10.1007/s00021-024-00904-0
Masakazu Yamamoto
{"title":"Time Evolution of the Navier–Stokes Flow in Far-Field","authors":"Masakazu Yamamoto","doi":"10.1007/s00021-024-00904-0","DOIUrl":"10.1007/s00021-024-00904-0","url":null,"abstract":"<div><p>Asymptotic expansion in far-field for the incompressible Navier–Stokes flow are established. It is well known that a velocity decays slowly in far-field. This property prevents classical procedure giving asymptotic expansions of solutions with high-order. In this paper, under natural settings and moment conditions on the initial vorticity, technique of renormalization together with Biot–Savart law derives an asymptotic expansion for velocity with high-order. Especially scalings and large-time behaviors of the expansions are clarified. By employing them, time evolution of velocity in far-field is drawn. As an appendix, asymptotic behavior of solutions as time variable tends to infinity is given. In this assertion, large-time behavior of velocity is discovered clearly.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142443176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Remark on the Local Well-Posedness of Compressible Non-Newtonian Fluids with Initial Vacuum 关于具有初始真空的可压缩非牛顿流体的局部良好拟合的备注
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-10-16 DOI: 10.1007/s00021-024-00901-3
Hind Al Baba, Bilal Al Taki, Amru Hussein
{"title":"Remark on the Local Well-Posedness of Compressible Non-Newtonian Fluids with Initial Vacuum","authors":"Hind Al Baba,&nbsp;Bilal Al Taki,&nbsp;Amru Hussein","doi":"10.1007/s00021-024-00901-3","DOIUrl":"10.1007/s00021-024-00901-3","url":null,"abstract":"<div><p>We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in https://doi.org/10.1007/s00208-021-02301-8 can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in https://doi.org/10.1016/j.matpur.2003.11.004 for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic <span>(W^{2,p})</span>-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-024-00901-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142443362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Isolated Singularities for the Stationary Navier–Stokes System 关于静态纳维-斯托克斯系统的孤立奇点
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-10-15 DOI: 10.1007/s00021-024-00905-z
Alfonsina Tartaglione
{"title":"On Isolated Singularities for the Stationary Navier–Stokes System","authors":"Alfonsina Tartaglione","doi":"10.1007/s00021-024-00905-z","DOIUrl":"10.1007/s00021-024-00905-z","url":null,"abstract":"<div><p>The classical problem of removable singularities is considered for solutions to the stationary Navier–Stokes system in dimension <span>(nge 3)</span> and an old theorem of Shapiro (TAMS 187:335–363, 1974) is recovered and extended to solutions in a half ball vanishing on the flat boundary. Moreover, for <span>(n=4)</span> it is proved that there are not distributional solutions, smooth away from the singularity and such that <span>(u(x)=O(|x|^{-1}))</span>.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-024-00905-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Liouville-Type Theorems for the Stationary Ideal Magnetohydrodynamics Equations in (textbf{R}^n) (textbf{R}^n) 中固定理想磁流体力学方程的利乌维尔式定理
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-10-10 DOI: 10.1007/s00021-024-00902-2
Lv Cai, Ning-An Lai, Anthony Suen, Manwai Yuen
{"title":"Liouville-Type Theorems for the Stationary Ideal Magnetohydrodynamics Equations in (textbf{R}^n)","authors":"Lv Cai,&nbsp;Ning-An Lai,&nbsp;Anthony Suen,&nbsp;Manwai Yuen","doi":"10.1007/s00021-024-00902-2","DOIUrl":"10.1007/s00021-024-00902-2","url":null,"abstract":"<div><p>In this paper, we establish Liouville-type theorems for the stationary ideal compressible magnetohydrodynamics system in <span>(textbf{R}^n)</span> with <span>(nin {1, 2, 3})</span>. We address various cases when the finite energy condition is in force and the stationary density function <span>(rho )</span> satisfies <span>(displaystyle lim _{|x|rightarrow infty }rho (x)=rho _infty ge 0)</span>. Our proof relies heavily on the good structure of the nonlinear magnetic force term and the usage of well-chosen smooth cut-off test functions.\u0000</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142411235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Non-zonal Rossby–Haurwitz Solutions of the 2D Euler Equations on a Rotating Ellipsoid 旋转椭球体上二维欧拉方程的非纵向罗斯比-豪尔维茨解
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-10-01 DOI: 10.1007/s00021-024-00884-1
Chenghao Xu
{"title":"The Non-zonal Rossby–Haurwitz Solutions of the 2D Euler Equations on a Rotating Ellipsoid","authors":"Chenghao Xu","doi":"10.1007/s00021-024-00884-1","DOIUrl":"10.1007/s00021-024-00884-1","url":null,"abstract":"<div><p>In this article, we investigate the incompressible 2D Euler equations on a rotating biaxial ellipsoid, which model the dynamics of the atmosphere of a Jovian planet. We study the non-zonal Rossby–Haurwitz solutions of the Euler equations on an ellipsoid, while previous works only considered the case of a sphere. Our main results include: the existence and uniqueness of the stationary Rossby–Haurwitz solutions; the construction of the traveling-wave solutions; and the demonstration of the Lyapunov instability of both the stationary and the traveling-wave solutions.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numerical Analysis for a Non-isothermal Incompressible Navier–Stokes–Allen–Cahn System 非等温可压缩 Navier-Stokes-Allen-Cahn 系统的数值分析
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-09-21 DOI: 10.1007/s00021-024-00898-9
Diego A. Rueda-Gómez, Elian E. Rueda-Fernández, Élder J. Villamizar-Roa
{"title":"Numerical Analysis for a Non-isothermal Incompressible Navier–Stokes–Allen–Cahn System","authors":"Diego A. Rueda-Gómez,&nbsp;Elian E. Rueda-Fernández,&nbsp;Élder J. Villamizar-Roa","doi":"10.1007/s00021-024-00898-9","DOIUrl":"10.1007/s00021-024-00898-9","url":null,"abstract":"<div><p>In this paper we develop the numerical analysis for a non-isothermal diffuse-interface model, in dimension <span>(N=2, 3,)</span> that describes the movement of a mixture of two incompressible viscous fluids. This model consists of modified Navier–Stokes equations coupled with a phase-field equation given by a convective Allen–Cahn equation, and energy transport equation for the temperature; which admits a dissipative energy inequality. We propose an energy stable numerical scheme based on the Finite Element Method, and we analyze optimal weak and strong error estimates, as well as convergence towards regular solutions. In order to construct the numerical scheme, we introduce two extra variables (given by the gradient of the temperature and the variation of the energy with respect to the phase-field function) which allows us to control the strong regularity required by the model, which is one of the main difficulties appearing from the numerical point of view. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed, energy stable and satisfies a set of uniform estimates which allow to analyze the convergence of the scheme. Finally, we present some numerical simulations to validate numerically our theoretical results.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-024-00898-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Injection of Fluid from a Slot into a Stream: Uniqueness 将流体从缝隙注入溪流：独特性
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-09-21 DOI: 10.1007/s00021-024-00896-x
Lili Du, Yuanhong Zhao
{"title":"Injection of Fluid from a Slot into a Stream: Uniqueness","authors":"Lili Du,&nbsp;Yuanhong Zhao","doi":"10.1007/s00021-024-00896-x","DOIUrl":"10.1007/s00021-024-00896-x","url":null,"abstract":"<div><p>This is a sequel work on the existence of the solution to the free boundary problem on injection of fluid from a slot into a uniform stream with two free boundaries by Stojanovic (IMA J Appl Math 41:237–253, 1988). However, the uniqueness of the solution to the two-phase fluids problem with two free boundaries remains unresolved. In this paper, we will establish the asymptotic behavior of the flow in the upstream and prove the uniqueness of the solution to this problem.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Uniform (L^p) Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing 非均质二维纳维-斯托克斯方程组解的均匀 $$L^p$$ 估计数及其在具有局部感应的趋化-流体系统中的应用
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-09-16 DOI: 10.1007/s00021-024-00899-8
Mario Fuest, Michael Winkler
{"title":"Uniform (L^p) Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing","authors":"Mario Fuest,&nbsp;Michael Winkler","doi":"10.1007/s00021-024-00899-8","DOIUrl":"10.1007/s00021-024-00899-8","url":null,"abstract":"<div><p>The chemotaxis-Navier–Stokes system </p><div><div><span>$$begin{aligned} left{ begin{array}{rcl} n_t+ucdot nabla n &amp; =&amp; Delta big (n c^{-alpha } big ), c_t+ ucdot nabla c &amp; =&amp; Delta c -nc, u_t + (ucdot nabla ) u &amp; =&amp; Delta u+nabla P + nnabla Phi , qquad nabla cdot u=0, end{array} right. end{aligned}$$</span></div></div><p>modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain <span>(Omega subset mathbb R^2)</span>. For all <span>(alpha &gt; 0)</span> and all sufficiently regular <span>(Phi )</span>, we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier–Stokes system. As a crucial new challenge, our analysis requires a priori estimates for <i>u</i> at a point in the proof when knowledge about <i>n</i> is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time <span>(L^p)</span> estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time <span>(L^2)</span> norm of the force term raised to an arbitrary small power.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-024-00899-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142264117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Self-Similar Solution of the Generalized Riemann Problem for Two-Dimensional Isothermal Euler Equations 二维等温欧拉方程广义黎曼问题的自相似解
IF 1.2 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2024-09-05 DOI: 10.1007/s00021-024-00897-w
Wancheng Sheng, Yang Zhou
{"title":"Self-Similar Solution of the Generalized Riemann Problem for Two-Dimensional Isothermal Euler Equations","authors":"Wancheng Sheng,&nbsp;Yang Zhou","doi":"10.1007/s00021-024-00897-w","DOIUrl":"10.1007/s00021-024-00897-w","url":null,"abstract":"<div><p>In this paper, a kind of classic generalized Riemann problem for 2-dimensional isothermal Euler equations for compressible gas dynamics is considered. The problem is the gas <span>((u_{0}, v_{0}, r_{0} mid x mid ^{beta }))</span> in the rectangular region expands into the vacuum. We construct the solution of the following form </p><div><div><span>$$begin{aligned} u=u(xi , eta ), v=v(xi , eta ), rho =t^{beta } varrho (xi , eta ), xi =frac{x}{t}, eta =frac{y}{t}, end{aligned}$$</span></div></div><p>where <span>(rho )</span> and (<i>u</i>, <i>v</i>) denote the density and the velocity fields respectively, and <span>(u_{0}, v_{0}, r_{0}&gt;0)</span> and <span>(beta in (-1,0) cup (0,+infty ))</span> are constants. The continuity of the self-similar solution depends on the value of <span>(beta )</span>. Under certain conditions, we get a weak solution with shock wave, which is necessarily generated initially and move apart along a plane. Furthermore, by the method of characteristic analysis, we explain the mechanism of the shock wave.\u0000</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 4","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","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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