Qionglei Chen, Changxing Miao, Zhifei Zhang
We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, i. e. $\sup_{j\in\Z}\int_0^T\|Δ_j(\na\times u)\|_\infty dt$, where $Δ_j$ is a frequency localization on $|ξ|\approx 2^j$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove a localized non blow-up theorem of the Beale-Kato-Majda type for the solution of the 3D incompressible Euler equations.
In this paper, we establish an analog of the Beale-Kato-Majda type criterion for singularities of smooth solutions of a hydrodynamic system modeling vesicle and fluid interactions. The result shows that the maximum norm of the vorticity alone controls the breakdown of smooth solutions.
In this paper, we prove a Beale--Kato--Majda blow-up criterion in terms of the gradient of the velocity only for the strong solution to the 3-D compressible nematic liquid crystal flows with nonnegative initial densities. More precisely, the strong solution exists globally if the $L^{1}(0,T;L^{\infty})$-norm of the gradient of the velocity $u$ is bounded. Our criterion improves the recent result of X. Liu and L. Liu (\cite{LL}, A blow-up criterion for the compressible liquid crystals system, arXiv:1011.4399v2 [math-ph] 23 Nov. 2010).
Beale--Kato--Majda-type continuation criteria for Hall- and electron-magnetohydrodynamics
该研究针对带电阻的电子MHD和Hall-MHD，分别建立了类似Beale--Kato--Majda结果的无量纲延拓准则，明确了对应解可延拓的积分条件。
We obtain a Beale-Kato-Majda-type criterion with optimal frequency and temporal localization for the 3D Navier-Stokes equations. Compared to previous results our condition only requires the control of Fourier modes below a critical frequency, whose value is explicit in terms of time scales. As applications it yields a strongly frequency-localized condition for regularity in the space $B^{-1}_{\infty,\infty}$ and also a lower bound on the decaying rate of $L^p$ norms $2\leq p <3$ for possible blowup solutions. The proof relies on new estimates for the cutoff dissipation and energy at small time…
We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier-Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is \textit{a priori} estimate for an important quantity under the assumption that the density is upper bounded, whose divergence can be viewed as the effective viscous flux.
We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions.
We consider a multidimensional Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of $\Rd$ we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.
We derive a criterion for the breakdown of solutions to the Oldroyd-B model in $\R^3$ in the limit of zero Reynolds number (creeping flow). If the initial stress field is in the Sobolev space $H^m$, $m> 5/2$, then either a unique solution exists within this space indefinitely, or, at the time where the solution breaks down, the time integral of the $L^\infty$-norm of the stress tensor must diverge. This result is analogous to the celebrated Beale-Kato-Majda breakdown criterion for the inviscid Eluer equations of incompressible fluids.
A Generalized Beale-Kato-Majda Breakdown Criterion for the free-boundary problem in Euler Equations with Surface Tension
本文针对带表面张力且具有自由表面边界的不可压缩 Euler 方程光滑解，提出了一个广义 Beale-Kato-Majda 破缺准则，并证明当自由边界固定时该准则可退化为 Ferrari 的经典结果。
We establish an analog of the Beale-Kato-Majda criterion for singularities of smooth solutions of the system of PDE arising in the Oldroyd model for ideal viscoelastic flow.
We study the regularity problem of a nematic liquid crystal model with local configuration represented by Q-tensor in three dimensions. It was an open question whether the classical Prodi-Serrin condition implies regularity for this model. Applying a wavenumber splitting method, we show that a solution does not blow-up under certain extended Beale-Kato-Majda condition solely imposed on velocity. This regularity criterion automatically implies that the classical Prodi-Serrin or Beale-Kato-Majda condition prevents blow-up of solutions.
Blow-Up Criteria and Weak--Strong Uniqueness for Compressible Fluid--Viscoelastic Shell Interactions
本文研究了压缩流体-粘弹性壳体相互作用系统强解的有限时间爆破准则及弱-强唯一性原理，通过高阶能量估计和Lipschitz型控制保证解的延展性。
In this paper, we consider the ideal magnetic Bénard problem in both two and three dimensions and prove local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}+1, n = 2,3$. In addition, a necessary condition is derived for singularity development with respect to the $BMO$-norm of the vorticity and electrical current, generalising the Beale-Kato-Majda condition for ideal hydrodynamics.
In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschtiz function and the free surface belongs to $C^{\f32+\varepsilon}$. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.
In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.
In this paper we study the magneto-micropolar fluid equations in $\R^3$, prove the existence of the strong solution with initial data in $H^s(\R^3)$ for $s> {3/2}$, and set up its blow-up criterion. The tool we mainly use is Littlewood-Paley decomposition, by which we obtain a Beale-Kato-Majda type blow-up criterion for smooth solution $(u,ω,b)$ which relies on the vorticity of velocity $\nabla\times u$ only.
In this paper, we prove the global well-posedness for the three-dimensional magnetohydrodynamics (MHD) equations with zero viscosity and axisymmetric initial data. First, we analyze the problem corresponding to the Sobolev regularities $ H^s\times H^{s-2}$, with $ s>5/2$. Second, we address the same problem but for the Besov critical regularities $ B_{p,1}^{3/p+1}\times B_{1,p}^{3/p-1}$, $2\leq p\leq \infty$. This case turns out to be more subtle as the Beale-Kato-Majda criterion is not known to be valid for rough regularities.
In this paper we deal with the existence of local strong solution for a perfect compressible viscous fluid, heat conductive and self gravitating, coupled with a first order kinetics used in astrophysical hydrodynamical models. In our setting the vacuum is allowed and as a byproduct of the existence result we get a blow-up criterion for the local strong solution. Moreover we prove a blow-up criterion for the local strong solutions in terms of the velocity gradient, the mass fraction gradient and the temperature similar to the well known Beale-Kato-Majda criterion for ideal incompressible flows.
This article consists of a detailed geometric study of the one-dimensional vorticity model equation $$ω_{t} + uω_{x} + 2ωu_{x} = 0, \qquad ω= H u_{x}, \qquad t\in\mathbb{R},\; x\in S^{1}\,,$$ which is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on $\operatorname{Diff}(S^{1})$ when the latter is endowed with the right-invariant homogeneous $\dot{H}^{1/2}$-metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Fu…
In this paper we establish local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}$ to the viscous, zero thermal-diffusive Boussinesq equations in $\mathbb{R}^n , n = 2,3$. Beale-Kato-Majda type blow-up criterion has been established in three-dimensions with respect to the $BMO$-norm of the vorticity. We further prove the local-in-time existence and blow-up criterion for non-viscous and fully ideal Boussinesq systems. Commutator estimates due to Kato and Ponce (1988) \cite {KP} and Fefferman et. al. (2014) \cite {Fe} play important roles in the calculations.
Blow-up of strong solutions of the Thermal Quasi-Geostrophic equation
本文针对热准地转方程，借鉴Beale-Kato-Majda准则，给出了其强解发生爆破的判据。
We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.
We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,α}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale-Kato-Majda criterion cannot be improved in the class of $C^{1,α}$ solutions.
Blow-up criterion to the compressible full Navier-Stokes equations with inflow-outflow boundary conditions
针对具有流入-流出边界条件的可压缩全Navier-Stokes方程，建立了强解存在性，并推导出三维情形下类似BKM准则的新正则性准则，改进了已有结果。
The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent approaches to the problem. We first review Kato's classical local well-posedness result in the Sobolev space and derive the celebrated Beale-Kato-Majda criterion for finite time blow-up. Then, we discuss recent refinements of the criterion as well as geometric type of theorems on the sufficiency condition for the regularity of solutions. After that we review resu…
We prove by an explicit construction that solutions to incompressible 3D Euler equations defined in the periodic cube can be mapped bijectively to a new system of equations whose solutions are globally regular. We establish that the usual Beale-Kato-Majda criterion for finite-time singularity (or blowup) of a solution to the 3D Euler system is equivalent to a condition on the corresponding \emph{regular} solution of the new system. In the hypothetical case of Euler finite-time singularity, we provide an explicit formula for the blowup time in terms of the regular solution of the new system. Th…
In this paper, we consider the $3$D compressible radiation hydrodynamic (RHD) equations with thermal conductivity in a bounded domain. The existence of unique local strong solutions is firstly established when the initial data are arbitrarily large and satisfy some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set. Moreover, we show that if the initial vacuum domain is not so irregular, then the compatibility condition is necessary and sufficient to guarantee the existence of the unique strong solution. Finally, …
We define a right-invariant Riemannian metric on the group of contactomorphisms and study its Euler-Arnold equation. If the metric is associated to the contact form, the Euler-Arnold equation reduces to $m_t + u(m) + (n+2) mE(f) = 0$, in terms of the Reeb field $E$, a stream function $f$, the contact vector field $u$ defined by $f$, and the momentum $m = f - Δf$. Here the equation is considered on a compact manifold $M$ of dimension $2n+1$. When $n=0$ this reduces to the Camassa-Holm equation, and we emphasize the analogy with the higher-order equation. We use the usual momentum conservation l…
A new blowup criterion for the 3D barotropic compressible Navier-Stokes equations with vacuum
该研究针对含真空的三维正压可压缩Navier-Stokes方程，在三类边界/初值问题下提出新的爆破准则，推广了经典的Beale-Kato-Majda准则，补充了已有相关研究结果。
Sharp non-uniqueness of weak solutions to 2D magnetohydrodynamic equations
本文证明二维粘性电阻磁流体动力学（MHD）方程在$L^2_t L^p(\mathbb{R}^2) \cap L^1_t W^{1,p}(\mathbb{R}^2)$中弱解非唯一，其中$1\le p<\infty$，展示了Ladyzhenskaya-Prodi-Serrin条件在端点$(2,\infty)$的尖锐性，且解位于Beale-Kato-Majda准则的边界上。
In this paper, we obtain a blow up criterion for strong solutions to the 3-D compressible Naveri-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow. The key ingredients in our analysis are the a priori super-norm estimate of the momentum by a Moser-iteration and an estimate of the space-time square mean of the gradient of the density. In addition, initial vacuum is allowed in our case.
Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension
研究具有表面张力的三维不可压缩欧拉方程在有界域中强解的爆破准则，在\(H^3\times H^4\)正则性水平上建立，不设多种假设，给出有限时间爆破时的四种机制，单连通域有细化，无旋流时准则简化。
Sharp Hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations
针对不可压缩自由边界欧拉方程，我们建立了Hadamard局部适定性、增强唯一性、稳定性界、能量估计及尖锐延拓准则，并给出正则解构造的新证明。
This work studies a simplified model of the gravitational instability of an initially homogeneous infinite medium, represented by $\TT^d$, based on the approximation that the mean fluid velocity is always proportional to the local acceleration. It is shown that, mathematically, this assumption leads to the restricted Patlak-Keller-Segel model considered by Jäger and Luckhaus or, equivalently, the Smoluchowski equation describing the motion of self-gravitating Brownian particles, coupled to the modified Newtonian potential that is appropriate for an infinite mass distribution. We discuss some o…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。