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Ill-Posedness of the Euler Equations Linearized around Homogeneous Steady States 齐次稳态下线性化欧拉方程的病态性
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-11-09 DOI: 10.1007/s00021-026-01074-x
Matei P. Coiculescu
{"title":"Ill-Posedness of the Euler Equations Linearized around Homogeneous Steady States","authors":"Matei P. Coiculescu","doi":"10.1007/s00021-026-01074-x","DOIUrl":"10.1007/s00021-026-01074-x","url":null,"abstract":"<div><p>Let <span>(L^2_m(mathbb {R}^2))</span> be the space of square-integrable functions on <span>(mathbb {R}^2)</span> with <i>m</i>-fold rotational symmetry. Let <span>(overline{omega }(r,theta ) = r^{-alpha }f(theta ))</span> be a homogeneous steady state of the two-dimensional incompressible Euler equations with <span>((mcdot l))</span>-fold rotational symmetry. If <span>(f(theta ))</span> is a constant function we say that <span>(overline{omega })</span> is a radial power-law vortex. We prove that the incompressible Euler equations in vorticity form, linearized around any homogeneous steady state <span>(overline{omega })</span> that is not a radial power-law vortex, are ill-posed on <span>(L^2_m(mathbb {R}^2))</span> for any <span>(mge 2)</span>, any <span>(lge 1)</span> and <span>(alpha in (0,1))</span>.\u0000</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"29 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149369250","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
(L^p) Estimates and the Vanishing Vertical Viscosity Limit for the 2D Anisotropic Navier–Stokes Equations with No-Slip Boundary Conditions (L^p) 具有无滑移边界条件的二维各向异性Navier-Stokes方程的估计和垂直粘度极限消失
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-11-02 DOI: 10.1007/s00021-026-01067-w
Chongsheng Cao, Yanqiu Guo
{"title":"(L^p) Estimates and the Vanishing Vertical Viscosity Limit for the 2D Anisotropic Navier–Stokes Equations with No-Slip Boundary Conditions","authors":"Chongsheng Cao,&nbsp;Yanqiu Guo","doi":"10.1007/s00021-026-01067-w","DOIUrl":"10.1007/s00021-026-01067-w","url":null,"abstract":"<div><p>This paper studies the vanishing vertical viscosity limit for the two-dimensional Navier–Stokes equations with anisotropic viscosity. The fluid domain is bounded above and below by impenetrable walls, with a <i>no-slip</i> boundary condition imposed on the bottom wall and a slip boundary condition imposed on the top wall. We derive global <span>(L^p)</span> energy estimates with explicit dependence on the horizontal and vertical viscosities. By introducing a decomposition of the pressure into a flow component and a boundary-layer component, we isolate and quantify boundary effects associated with the vertical viscosity. Using these estimates, we prove that solutions converge strongly in <span>(L^p(Omega ))</span>, for any <span>(2 le p &lt; infty )</span>, to solutions of the horizontally viscous limit system as the vertical viscosity tends to zero, provided that the initial data are sufficiently regular. The main difficulty arises from the mismatch in boundary conditions, namely, the no-slip condition in the original problem versus the slip condition in the limiting system. The purpose of this paper is to understand the associated boundary layer phenomena under stronger assumptions. Our model captures the essential features of the boundary layer behavior.\u0000</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-026-01067-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149338096","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
Vorticity Confinement for 2D Incompressible Flows in an Infinite Cylinder 无限大圆柱体中二维不可压缩流动的涡量约束
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-10-28 DOI: 10.1007/s00021-026-01068-9
Paolo Buttà, Guido Cavallaro
{"title":"Vorticity Confinement for 2D Incompressible Flows in an Infinite Cylinder","authors":"Paolo Buttà,&nbsp;Guido Cavallaro","doi":"10.1007/s00021-026-01068-9","DOIUrl":"10.1007/s00021-026-01068-9","url":null,"abstract":"<div><p>We study the confinement of vorticity for two-dimensional incompressible flows in an infinite cylinder. For Navier–Stokes solutions with non-negative and compactly supported initial vorticity, we derive quantitative decay estimates showing that the vorticity mass outside regions whose distance from the initial support grows like <span>(sqrt{tlog ^alpha t})</span> (with <span>(alpha &gt;1)</span>) or like <span>(t^beta )</span> (with <span>(beta &gt;1/2)</span>) becomes, respectively, super-polynomially or stretched-exponentially small. The analysis combines an iterative scheme with an antisymmetry property of the Biot-Savart kernel. In the Euler case, by coupling this approach with a first-moment estimate from [6], we recover the confinement bound of [6] and refine it slightly: the diameter of the vorticity support grows at most like <span>((tlog t)^{1/3})</span>, rather than <span>(t^{1/3}log ^2 t)</span>.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149339016","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
Invariant Geometric Properties in the Dynamics of Bounded Planar Hele-Shaw Flows 有界平面Hele-Shaw流动力学中的不变几何性质
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-10-26 DOI: 10.1007/s00021-026-01065-y
Mihai Aron
{"title":"Invariant Geometric Properties in the Dynamics of Bounded Planar Hele-Shaw Flows","authors":"Mihai Aron","doi":"10.1007/s00021-026-01065-y","DOIUrl":"10.1007/s00021-026-01065-y","url":null,"abstract":"<div><p>The geometric property of <span>(sigma )</span>-superstarlikeness, along with its related notions–<span>(sigma )</span>-superstarlikeness of order <span>(alpha )</span> and <span>(sigma )</span>-superstarlikeness of complex order–are introduced and studied in this paper. These concepts are shown not only to cover but also to generalize a wide range of known time-invariant geometric properties of the moving boundary in the inner Hele-Shaw flow problem for bounded domains, i.e., two-dimensional bounded viscous fluid flows under injection in the absence of surface tension. We prove that, under suitable parameter constraints, these properties are preserved in time. Consequently, this work extends, unifies, and completes recent research on the Hele-Shaw flow problem in the absence of surface tension.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-026-01065-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149339411","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
Beale–Kato–Majda-Type Continuation Criteria for Hall- and Electron-Magnetohydrodynamics 霍尔和电子磁流体力学的beale - kato - majda型延拓判据
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-10-21 DOI: 10.1007/s00021-026-01066-x
Mimi Dai, Sung-Jin Oh
{"title":"Beale–Kato–Majda-Type Continuation Criteria for Hall- and Electron-Magnetohydrodynamics","authors":"Mimi Dai,&nbsp;Sung-Jin Oh","doi":"10.1007/s00021-026-01066-x","DOIUrl":"10.1007/s00021-026-01066-x","url":null,"abstract":"<div><p>We show that regular solutions to electron-MHD with resistivity can be continued as long as the time integral of the supremum of the current gradient remains finite. This dimensionless continuation criterion is a Beale–Kato–Majda (BKM) type of condition. A similar continuation criterion, formulated in terms of the time integral of the supremum of the vorticity, velocity gradient and current gradient, is established for the Hall-MHD with resistivity as well. We emphasize that the BKM type criterion for electron/Hall-MHD was not previously known, and such criterion for the electron/Hall-MHD without resistivity still remains open. The absence of the classical BKM criterion for the non-resistive electron/Hall-MHD is an additional evidence of the very singular nature of the Hall term, besides what have been discovered on strong ill-posedness of these systems, for instance see [5, 10, 12].</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-026-01066-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149234681","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
Error Estimates for a Finite Element Approximation of a Generalized Viscoelastic Flow Model 广义粘弹性流动模型有限元近似的误差估计
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-10-14 DOI: 10.1007/s00021-026-01063-0
Yingwen Guo, Mengmeng Liu, Wenlin Qiu, Xiangcheng Zheng
{"title":"Error Estimates for a Finite Element Approximation of a Generalized Viscoelastic Flow Model","authors":"Yingwen Guo,&nbsp;Mengmeng Liu,&nbsp;Wenlin Qiu,&nbsp;Xiangcheng Zheng","doi":"10.1007/s00021-026-01063-0","DOIUrl":"10.1007/s00021-026-01063-0","url":null,"abstract":"<div><p>This work analyzes the finite element approximation to a viscoelastic flow model, which generalizes the Navier–Stokes equation and Oldroyd’s model by introducing the tempered power-law memory kernel. We prove regularity and long-time exponential decay of the solutions, as well as a long-time convolution-type Grönwall inequality to support numerical analysis. A Volterra–Stokes projection is developed and analyzed to facilitate the parabolic-type duality argument, leading to the long-time error estimates and exponential decay of velocity and pressure. A benchmark problem of planar four-to-one contraction flow is simulated to substantiate the generality of the proposed model in comparison with the Navier–Stokes equation and Oldroyd’s model.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149189064","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
Optimal Decay Rate to the Viscous Contact Discontinuity for the System of Heat-Conductive Ideal Gas Without Viscosity 无粘性导热理想气体系统粘性接触不连续的最佳衰减率
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-10-08 DOI: 10.1007/s00021-026-01062-1
Guiqiong Gong, Lin He
{"title":"Optimal Decay Rate to the Viscous Contact Discontinuity for the System of Heat-Conductive Ideal Gas Without Viscosity","authors":"Guiqiong Gong,&nbsp;Lin He","doi":"10.1007/s00021-026-01062-1","DOIUrl":"10.1007/s00021-026-01062-1","url":null,"abstract":"<div><p>This paper is concerned with the time-decay rates of the viscous contact wave in 1D heat-conductive ideal gas without viscosity. For the case without the zero mass condition, although the heat-conductive ideal gas without viscosity studied in this paper lacks dissipation in terms of velocity, we can still prove the same optimal decay rate as the compressible Navier–Stokes equation. Precisely, we can obtain the optimal decay rate <span>((1+t)^{-1/2})</span> for the perturbation in <span>(L^{infty })</span>-norm. The crucial point of the analysis is as follows. We refine the estimates for both anti-derivatives and the original perturbations in the first, and then, we use an innovative transformation to ensure the structural conditions continue to hold for the system of derivatives. With this approach, we obtain the optimal decay rate.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149132503","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
On the Exchange of Stability for the Subcritical Laminar Flow 亚临界层流的稳定性交换
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-10-06 DOI: 10.1007/s00021-026-01059-w
Vladimir Kozlov, Oleg Motygin
{"title":"On the Exchange of Stability for the Subcritical Laminar Flow","authors":"Vladimir Kozlov,&nbsp;Oleg Motygin","doi":"10.1007/s00021-026-01059-w","DOIUrl":"10.1007/s00021-026-01059-w","url":null,"abstract":"<div><p>We consider steady water waves in a two-dimensional channel bounded below by a flat, rigid bottom and above by a free surface. Surface tension is neglected, and the flow is rotational with constant vorticity <i>a</i>. We analyze an analytic branch of Stokes waves bifurcating from a subcritical laminar flow, with the wave period serving as the bifurcation parameter. Along this branch, the first eigenvalue of the Fréchet derivative remains negative. Our main focus is the second eigenvalue; its sign plays a crucial role in the analysis of subharmonic bifurcations. This small eigenvalue determines the validity of the principle of exchange of stabilities: a positive sign confirms it, while a negative sign indicates its violation. Furthermore, a positive second eigenvalue corresponds to an increasing period along the bifurcation curve near the critical point, whereas a negative sign implies period decrease. We investigate how the sign of the second eigenvalue depends on the Bernoulli constant <i>R</i> (equivalently, the laminar flow depth <i>d</i>) and the vorticity <i>a</i>. We show that for each <i>a</i> there exists a critical depth <span>(d_0(a))</span> such that the second eigenvalue is positive for <span>(d&lt;d_0(a))</span> and negative for <span>(d&gt;d_0(a))</span>. In the laminar flow, a stagnation point forms when the depth exceeds a threshold <span>(d_{text {s}}(a))</span>. We demonstrate that <span>(d_0(a) &lt; d_{text {s}}(a))</span> for <span>(a &gt; a_0 approx -1.01803)</span>, whereas <span>(d_0(a) &gt; d_{text {s}}(a))</span> for <span>(a &lt; a_0)</span>. We also verify the property of formal stability by a description of the domain in (<i>a</i>, <i>d</i>) variables, where this property holds. Numerical illustrations of these properties are presented in the paper.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149132420","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
Local-in-Time Existence of Strong Solutions to a Class of Compressible Power-Law Flows 一类可压缩幂律流强解的局部存在性
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-09-29 DOI: 10.1007/s00021-026-01058-x
Fang Li, Chang Mengge, Guo Zhenhua
{"title":"Local-in-Time Existence of Strong Solutions to a Class of Compressible Power-Law Flows","authors":"Fang Li,&nbsp;Chang Mengge,&nbsp;Guo Zhenhua","doi":"10.1007/s00021-026-01058-x","DOIUrl":"10.1007/s00021-026-01058-x","url":null,"abstract":"<div><p>We consider a model of the compressible non-Newtonian fluids for power-law flow fulfilling a periodic domain in <span>({mathbb R}^3,)</span> in which the extra stress tensor is induced by a potential with <i>p</i>(<i>t</i>, <i>x</i>)-structure. The local-in-time existence and uniqueness of strong solutions is proved for all <span>(frac{7}{5} &lt; inf p(t,x) leqslant sup p(t,x) leqslant 2.)</span> Further, two improved blow-up criterions for the obtained strong solution is given.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149133201","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
On Bridging Analyticity and Sparseness in Hyperdissipative Navier–Stokes Systems 超耗散Navier-Stokes系统的桥接解析性和稀疏性
IF 1.5 3区 数学
Journal of Mathematical Fluid Mechanics Pub Date : 2026-09-28 DOI: 10.1007/s00021-026-01054-1
Moses Patson Phiri
{"title":"On Bridging Analyticity and Sparseness in Hyperdissipative Navier–Stokes Systems","authors":"Moses Patson Phiri","doi":"10.1007/s00021-026-01054-1","DOIUrl":"10.1007/s00021-026-01054-1","url":null,"abstract":"<div><p>We study the three-dimensional hyperdissipative Navier–Stokes system in the super-critical regime (below the Lions threshold). Leveraging the analyticity–sparseness gap in the case where the ratios of the lower-order derivatives with respect to the higher-order derivatives are bounded by constants, we introduce time-weighted bridge inequalities quantifying these ratios across all derivative levels. On one hand, the time weights are chosen in a way that the resulting lower bound on the radius of spatial analyticity still dominates the scale of sparseness of the super-level sets indicating that the flow entered the dissipation regime. On the other hand, they provide a less restrictive environment for the ratios giving the construction more flexibility. As an illustration of its utility, it is shown that – together with a harmonic-measure contraction on one-dimensional sparse sets – this mechanism rules out a more general class of the blow-up scenarios compared to the previous work by Farhat and Grujić (essentially a perturbation of the multidimensional geometric series).</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"28 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2026-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-026-01054-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149133094","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
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