Ruslan Salimov, Mariia Stefanchuk
In this paper, the estimate for growth of homeomorphic solutions of the Beltrami equation at infinity is obtained, provided that the dilatation quotient has a global finite mean oscillation.
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Strong Beltrami fields have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field itself times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) …
Global bifurcation of doubly periodic gravity-capillary waves on Beltrami flows
研究证明Beltrami流上存在全局稳定双周期重力-毛细波。核心方法是将稳态水波问题转化为特定分岔问题并应用全局分岔论证。主要贡献是首个真正三维表面波严格存在性结果。
Asymptotic behavior of solutions of the nonlinear Beltrami equation with the Jacobian
本文考察右端含 Jacobian 的非线性 Beltrami 方程正则同胚解在无穷远处的渐近行为，并通过多个例子证明所得界是尖锐的。
Localization of Beltrami fields: global smooth solutions and vortex reconnection for the Navier-Stokes equations
本研究构造了贝尔特拉米场局部化得到的无散向量场，既用于构建三维Navier-Stokes方程临界空间大初值下的全局光滑解，也借助双曲临界点这一拓扑不变量给出了全空间上的涡重联解析示例。
Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale
本文研究平面区域上拟共形映射在Sobolev、Bessel位势和Triebel-Lizorkin尺度下的全局正则性，利用边界几何和Beltrami系数光滑性建立最优条件，并证明在光滑性低于1的Triebel-Lizorkin类中该条件同样保证主解的全局正则性。
Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation
本文展示刚性变量椭圆结构可通过算术转换将韦卡方程转换为标准形式，无需求解偏微分方程。
Asymptotic behavior of $L^2$-normalized eigenfunctions of the Laplace-Beltrami operator on a closed Riemannian manifold
设$e(x,y,ł)$是谱函数，$χ_ł$是单位带谱投影算子，关于闭黎曼流形$M$上的Laplace-Beltrami算子$\D_M$。本文首先回顾了H{\”o}rmander（1968）给出的$e(x,x,ł)$作为$ł\to\infty$的一项渐近公式和作者（2004）给出的$\p^\al_x\p^\bt_y e(x,y,ł)|_{x=y}$作为$ł\to\infty$的一项渐近公式，以及由此得到的映射范数$\|χ_ł\|_{L_2\to L_p}$的精确渐近估计（$2\leq p\leq\infty$）（1988 $&$ 1989）和作者（2004）的映射规范$\|χ_ł\|_{L_2\to {\rm Sobolev} L_p}$。本文证明了当$x$与$y$之间的黎曼距离为${\rm O}(1/ł)$时，$e(x,y,ł)$的一项渐近公式为$ł\to\infty$.作为结果，我们得到了映射范数$\|χ_ł\|_{L_2\to C^\d}$（$0<\d<1$）的精确估计，其中$C^\d(M)$是$M$上指数为$\d$的H{\”o}lder连续函数空间.此外，我们显示的本征函数$e_ł$的几何性质：$\D_M e_ł+ł^2 e_ł=0$，它说，$1/ł$是相当于$e_ł$的节点集（$e_ł$消失）和$e_ł$的集中集（$e_ł$达到其最大值或最小值）之间的距离为$ł\to\in…
Bicomplex Hardy Classes of Solutions to Beltrami Equations and the Schwarz Boundary Value Problem
该研究定义了复单位圆盘上双复变Beltrami方程及相关方程解的双复值Hardy类，证明其具有经典全纯Hardy空间的边界行为，推广了已有结果并证明相关Schwarz、Dirichlet边值问题可解且给出解式。
We study the global-in-time Strichartz estimates for the Schrödinger equation on a class of scattering manifolds $X^{\circ}$. Let $\mathcal{L}_V=Δ_g+V$ where $Δ_g$ is the Beltrami-Laplace operator on the scattering manifold and $V$ is a real potential function on this setting. We first extend the global-in-time Strichartz estimate in Hassell-Zhang \cite{HZ} on the requirement of $V(z)=O(\langle z\rangle^{-3})$ to $O(\langle z\rangle^{-2})$ and secondly generalize the result to the scattering manifold with a mild trapped set as well as Bouclet-Mizutani\cite{BM} but with a potential. We also obt…
We study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on Damek-Ricci spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain global well-posedness results for the nonlinear wave equation.
In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely $e^{-itΛ_g^σ}$ with $σ\in (0,\infty)\backslash \{1\}$ and $Λ_g:=\sqrt{-Δ_g}$ where $Δ_g$ is the Laplace-Beltrami operator on asymptotically Euclidean manifolds $(\mathbb{R}^d,g)$. Let $f_0\in C^\infty_0(\mathbb{R})$ be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part $(1-f_0)(P)e^{-itΛ_g^σ}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 2$ inside a compact set under non-trapping condition. On the other hand, under t…
Geometrically distinct unbounded continua of solutions for Yamabe-type equations
本文研究闭黎曼流形上Yamabe型方程的全局分歧，在对称性和适当指数条件下，证明了从正特征值分歧出两两不相交的无界变号解连续统，并在球面及射影空间上给出几何相异连续统数量的下界。
In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form $Δu + W \cdot \nabla u = 0$ in $\mathbb{R}^2$, where $W = W_1 + i W_2$ with each $W_j$ real-valued. Under the assumptions that $W_j \in L^{q_j}$ for some $q_1 \in [2, \infty]$, $q_2 \in (2, \infty]$, and $W_2$ exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of…
Long-time decay estimates for the Schrödinger equation on manifolds
在本文中，我们发展了一个定量版本的恩斯的方法，建立全球的时间衰减估计的解决方案，薛定谔方程的流形。为了简化说明，我们将只考虑形式为$H := - {1/2} Δ_M$的哈密顿算子，其中$Δ_M$是流形$M$上的拉普拉斯-贝尔特拉米算子，该流形是服从非捕获条件的三维欧几里得空间$\R^3$的光滑紧扰动。我们建立了薛定谔方程$u_t = -iHu$的全局时间局部平滑估计。这里的主要新颖之处是全球的时间方面的估计，这迫使一个更详细的分析的低频和中频的演变比本地的时间理论。特别是，为了处理中频，我们需要一个量子定理（它反映了$H$没有嵌入本征值的事实），以及一个将解渐近分解为输入和输出分量的Enss方法的定量版本，而为了处理低频，我们需要一个庞加莱型不等式（它反映了$H$在零处没有本征函数或共振的事实）。
We consider the Klein--Gordon equation associated with the Laplace--Beltrami operator $Δ$ on real hyperbolic spaces of dimension $n\!\ge\!2$; as $Δ$ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well--posedness results for the corresponding semilinear equation with low regularity data.
Regularity results for elliptic equations on cones
该研究针对N维欧氏空间球扇形上的椭圆问题，证明了泊松方程解梯度有界的最优特征值条件，还得到了p-拉普拉斯方程解的加权全局利普希茨连续性与二阶正则性。
In this paper we investigate the large time behavior of the global weak entropy solutions to the symmetric Keyftiz-Kranzer system with linear damping. It is proved that as t tends to infinite the entropy solutions tend to zero in the L p norm
The three-dimensional equations for the compressible flow of liquid crystals are considered. An initial-boundary value problem is studied in a bounded domain with large data. The existence and large-time behavior of a global weak solution are established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent $γ>\frac32$.
We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$ as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere.
Asymptotic Behaviors of Global Solutions to Fourth-order Parabolic and Hyperbolic Equations with Dirichlet Boundary Conditions
本文研究了四阶抛物型和双曲型方程在狄利克雷边界条件下的全局解渐进行为，建立了解趋于平衡态的收敛性及收敛速率估计，并通过数值模拟验证结果。
We study the behavior at infinity in time of the global solution of the anisotropic quasi-geostrophic equation $θ\in C_b(\mathbb{R}^+,H^s( \mathbb{R}^2))$. We prove that this solution decays to zero as time goes to infinity in $L^p(\mathbb{R}^2)$, $p\in [2,+\infty],$ moreover, we prove also that $\lim\limits_{t\rightarrow +\infty}\|θ(t)\|_{H^s}=0$.
We consider the weak solution of the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in $\mathbb R^3$. When the initial data is small in $L^2$ and initial density is positive and essentially bounded, we first prove the existence of a global weak solution in $\mathbb R^3$. The large-time behavior of a global weak solution is also established.
Global existence and long term behavior of 2d electro-hydrodynamics
研究了有界区域上二维不可压缩牛顿流体与色散抛物椭圆方程组的耦合方程。对于L^2初始数据，证明了在时间上全局弱解的存在性，并以一定的速率收敛到平稳解。本文将Debye-Huckel系统的一系列工作推广和改进到流体力学的情况。
In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as $t\to \infty$.
Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology
本文主要研究二维欧几里得空间有界区域上关于未知函数u和w的抛物退化系统。该系统表现为某些生物过程的数学模型。证明了非负经典Hölder连续解的整体存在唯一性。文章的最后一部分研究了解的渐近性态。
In this paper, a parabolic type Kirchhoff equation and its stationary counterpart are considered. For the evolution problem, the precise decay rates of the weak solution and of the corresponding energy functional are derived. For the stationary problem, a ground-state solution is obtained by applying Lagrange multiplier method. Moreover, the asymptotic behaviors of the general global solutions are also described. These results extend some recent ones obtained in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers …
We consider a reaction-diffusion system which may serve as a model for a ferment catalytic reaction in chemistry. The model consists of a system of reaction diffusion equations with unbounded time dependent coefficients and different polynomial reaction terms. An exponential decay of the globally bounded solutions is proved. The key tool of the proofs are properties of analytic semigroups and some inequalities.
We are concerned with the $3$D-Navier-Stokes equations with Coriolis force. Existence and uniqueness of global solutions in homogeneous Besov spaces are obtained for large speed of rotation. In the critical case of the regularity, we consider a suitable initial data class whose definition is based on the Stokes-Coriolis semigroup and Besov spaces. Moreover, we analyze the asymptotic behavior of solutions in that setting as the speed of rotation goes to infinity.
Global well-posedness and long time behavior of 2D MHD equations with partial dissipation in half space
研究半空间中带部分耗散的二维MHD方程，利用预解式估计建立低阶整体适定性与解的长时间行为，并揭示磁场对导电流体的稳定效应。
This paper is concerned with time global behavior of solutions to nonlinear Schrödinger equation with a non-vanishing condition at the spatial infinity. Under a non-vanishing condition, it would be expected that the behavior is determined by the shape of the nonlinear term around the non-vanishing state. To observe this phenomenon, we introduce a generalized version of the Gross-Pitaevskii equation, which is a typical equation involving a non-vanishing condition, by modifying the shape of nonlinearity around the non-vanishing state. It turns out that, if the nonlinearity decays fast as a solut…
Asymptotic behavior of global solutions of the $u_t=Δu + u^{p}$
研究了半线性抛物问题{u_t=Δu + u^{p}，x \in \mathbb{R}^{N}，t>0 u（0）= u_{0}，x \in \mathbb{R}^{N}，t=0非负解的渐近性态.对于$1<p\leq 1+ 2/N$，该问题的非负解$u(t)$在有限时间内爆破.此外，如果$p> 1+ 2/N$和$u_{0}$的范数足够小，则该问题有全局解.在这项工作中，我们使用熵的方法来获得整体解$u(t)$的衰减率。
Global solutions and Asymptotic Behavior to a Norm-preserving Non-local Parabolic Flow
针对通过非局部项迫使解保持在L²球面上的非线性抛物模型，研究其在有界区域和欧几里得空间能量空间的局部与整体适定性，证明有界区域正初始条件下解强收敛于稳态且渐近收敛于基态。
In this paper the long time behavior of the micropolar fluid equations energy on three dimensional space are studied. We show that $ \| (u,w)(\cdot,t) \|_{{L^{2}(\mathbb{R}^{3})}} \to 0 $ as $t \to \infty$ for Leray-Hopf's global weak solutions in inviscid vortex case. Moreover, when the vortex viscosity are considered, i.e., $χ>0$, we obtain a (faster) decay for micro-rotational field: $ \| w (\cdot,t) \|_{{L^{2}(\mathbb{R}^{3})}} = o(t^{-1/2})$.
In this work, we study global existence, eventual smoothness and asymptotical behavior of positive solutions for a two-species chemotaxis consumption model in a bounded smooth but not necessarily convex domain $Ω\subset \mathbb{R}^n (n=2,3,4,5)$ with nonnegative initial data and homogeneous Neumann boundary data Under a smallness condition, boundedness of classical solutions and stabilization to constant equilibrium is known. Here, without any smallness condition, we show global existence and uniform-in-time boundedness of classical solutions in 2D and global existence, eventual smoothness and…
New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution $(u, θ) \in D(A)\times H^1$ improves considerably the main result of the recent article [7]. Our second result on global regularity of the solution $(u, θ) \in V \times H^1$ for both bounded domain and the whole space ${\mathbb R}^2$ is a new one. It has been open and also seems much more challenging than the fi…
This paper is concerned with the Cauchy problem of the two-dimensional MHD system with magnetic diffusion. It was proved that the MHD equations have a unique global strong solution around the equilibrium state $(0, e_1)$. Furthermore, the $L^2$ decay rate of the velocity and magnetic field is obtained.
We consider the Cauchy problem for one-dimensional (1D) barotropic compressible Navier-Stokes equations with density-dependent viscosity and large external force. Under a general assumption on the density-dependent viscosity, we prove that the Cauchy problem admits a unique global strong (classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently. As a consequence, we obtain the large time behavior of the solution without external forces.
We study the global existence of a unique strong solution and its large-time behavior of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations through a drag forcing term. The coupled system can be derived as the hydrodynamic limit of the Vlasov-Fokker-Planck/isentropic Navier-Stokes equations with strong local alignment forces. When the initial data is sufficiently small and regular, we establish the unique existence of the global $H^s$-solutions in a perturbation framework. We also provide the large-tim…
Global solution and long-time behavior for a problem of phase segregation of the Allen-Cahn type
本文研究了一个由偏微分方程和常微分方程组成的相分离模型。通过仔细定义后一方程的最大解，该方程组可化为一个带记忆项的Allen-Cahn方程。证明了光滑解的整体存在唯一性，并给出了Ω极限集的一个特征。
We consider a class of quasilinear wave equations in $3+1$ space-time dimensions that satisfy the "weak null condition" as defined by Lindblad and Rodnianski \cite{LR1}, and study the large time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is $-\partial_t^2 u + (1+u) Δu = 0$. Global solutions for such equations have been constructed by Lindblad \cite{Lindblad1,Lindblad2} and Alinhac \cite{Alinhac1}. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solution…
We consider generalized models on coral broadcast spawning phenomena involving diffusion, advection, chemotaxis, and reactions when egg and sperm densities are different. We prove the global-in-time existence of the regular solutions of the models as well as their temporal decays in two and three dimensions. We also show that the total masses of egg and sperm density have positive lower bounds as time tends to infinity in three dimensions.
Global well-posedness and large-time behavior for three-dimensional magneto-micropolar equations with horizontal dissipation
本文针对三维水平耗散磁微极方程，在初始数据及其垂直导数L²小性条件下建立整体适定性，得到解H¹范数最优衰减率，证明依赖方程特殊结构与各向异性索伯列夫型不等式。
We study the isentropic Euler equations with time-dependent damping, given by $\fracμ{(1+t)^λ}ρu$. Here, $λ,μ$ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when $0<λ<1,0<μ$ in multi-dimensions $n\geq 1$. The asymptotic behavior will coincide with the one that obtained by many authors in the case $λ=0$. We will also show that the solution can only decay polynomially in time while in the three dimensions, the vorticity will decay exponentia…
In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker sufficient condition for the small data global existence, and we investigate the asymptotic pointwise behavior of global solutions for systems satisfying this condition. As an application, the asymptotic behavior of global solutions under the Alinhac condition is also derived.
In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation $$ \partial_tv-b\mathcal{H}\partial^2_xv- aε\partial_x^3v=cv\partial_xv-dε\partial_x(v\mathcal{H}\partial_xv+\mathcal{H}(v\partial_xv)), $$ is globally well-posed in the energy space $H^1(\mathbb R)$. Moreover, we study the limit behavior when the small positive parameter $ε$ tends to zero and show that, under a condition on the coefficients $a$, $b$, $c$ and $d$, the solution $v_ε$ to this equation converges to the corresponding solution of the Benjamin-Ono equation.
In this paper, we consider a two-phase flow model consisting of the compressible Navier-Stokes systems with degenerate viscosity coupled with the compressible Navier-Stokes systems with constant viscosities via a drag force, which can be derived from Chapman-Enskog expansion for the compressible Navier-Stokes-Vlasov-Fokker-Planck system. For general initial data, we establish the global existence of weak solutions with finite energy to the initial value problem in the three-dimensional periodic domain, and prove the convergence of global weak solutions to its equilibrium state as the time tend…
In this paper, we study the three-dimensional non-isentropic compressible fluid-particle flows. The system involves coupling between the Vlasov-Fokker-Planck equation and the non-isentropic compressible Navier-Stokes equations through momentum and energy exchanges. For the initial data near the given equilibrium we prove the global well-posedness of strong solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential. %The proof is based on …
In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg- de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global $L^2 $ a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially
We study linear pseudoparabolic equations with unbounded and time-dependent coefficients. We solve the case which has remained open in several recent studies of pseudoparabolic equations with unbounded and time-dependent coefficients. In this work we get a sharp condition for the existence or non-existence of solutions. Conditions on the initial function and coefficient are provided so that every nontrivial positive solution blows up instantaneously. When the coefficient and the initial function do not grow too rapidly, we establish the existence and uniqueness of global solutions, for both ti…
Global boundedness and asymptotic behavior of the chemotaxis system for Alopecia Areata with weakly singular sensitivity
本文研究了带有弱奇异敏感性的银屑病趋化模型的齐次Neumann初始-边界值问题，证明了二维情况下经典解的全局有界性，并通过构造Lyapunov函数证明解以指数速度收敛到常数稳态。
The purpose of this work is to study the global wellposedness and large time behavior results of strong solutions for the compressible Oldroyd-B model derived by Barrett, Lu, Süli (Commun. Math. Sci., 15, 1265--1323, 2017). Exploiting the Harmonic analysis tools (especially Littlewood-Paley theory), we first study the global well-posedness of the model with small initial data in spaces with low regularity. Then, under a suitable condition involving only the low frequency of the initial data, we also obtain the optimal decay rates of the solutions. Compared with the result by Wang and Wen (Math…
Global existence and large time behavior for primitive equations with free boundary
本文研究带自由边界的三维原始方程，证明了强解的全局存在唯一性，且解会以指数或代数速率长时间收敛到平衡态，为海洋大气大尺度运动模拟提供理论支撑。
We consider the focusing mass-supercritical and energy-subcritical nonlinear Schrödinger equation (NLS). We are interested in the global behavior of the solutions to (NLS) with group invariance. By the group invariance, we can determine the global behavior of the solutions above the ground state standing waves.
Global existence and asymptotic behavior of the Westervelt--hyperbolic Pennes system
本文研究 Westervelt–Pennes–Cattaneo 非线性超声加热耦合模型，利用连续性论证与插值不等式证明低阶初始范数小、高阶范数任意大时解整体存在，并证明其指数衰减。
In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $\mathbb{R}^{1+d}$ with $d\geq 3$. We prove that in the energy space the solution verifies the integrated local energy decay estimates for the full range of energy subcritical and critical power. For the case when $p>1+\frac{2}{d-1}$, we derive a uniform weighted energy bound for the solution as well as inverse polynomial decay of the energy flux through hypersurfaces away from the light cone. As a consequence, the solution scatters in the energy space and in the critical Sobolev space fo…
In this paper, we study global existence and large time behaviors of strong solutions to the kinetic Cucker--Smale model coupled with the three dimensional incompressible Navier--Stokes equations in the whole space. Using the maximal regularity estimate on the Stokes equations, global-in-time strong solutions to the Cauchy problem of the coupled system are obtained under small initial data regime. The optimal decay rate of the system, in the sense that the energy of the system decays at a rate coinciding with the one corresponding to the underling incompressible Navier--Stokes equations withou…
Since the pioneering works by Aronson & Bénilan [C. R. Acad. Sci. Paris Sér., 1979] and Bénilan & Crandall [Johns Hopkins Univ. Press, 1981], it is well-known that first-order evolution problems governed by a nonlinear but homogeneous operator admit the smoothing effect that every corresponding mild solution is Lipschitz continuous at every positive time. Moreover, if the underlying Banach space has the Radon-Nikodým property, then these mild solution is a.e. differentiable, and the time-derivative satisfies global and point-wise bounds. In this paper, we show that these results remain true if…
In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: $ (|u'|^{l}u')' + c|u'|^αu' + d|u|^βu=0$, where $α,β,l,c, d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.
A global stability result for incompressible magnetohydrodynamics
本文研究了不可压缩磁流体动力学方程在任意维度上的全局稳定性，提出了一种基于高阶Sobolev范数的显式估计方法，并引入了Beltrami型初始数据以证明全局解的稳定性。
In this paper, we construct a new class of blowup solutions with elementary functions to the 3-dimensional compressible or incompressible Euler and Navier-Stokes equations. In detail, we obtain a class of global rotational exact solutions for the compressible fluids with $γ>1$:%} [c]{c}% ρ=\max\{\frac{γ-1}{Kγ}[ C^{2}[ x^{2}% +y^{2}+z^{2}-(xy+yz+xz)] -\dot{a}(t)(x+y+z)+b(t)], 0\} ^{\frac{1}{γ-1}} u_{1}=a(t)+C(y-z) u_{2}=a(t)+C(-x+z) u_{3}=a(t)+C(x-y). where a(t)=c_{0}+c_{1}t and b(t)=3c_{0}c_{1}t+{3/2}c_{1}^{2}t^{2}+c_{2}% with $C$, $c_{0}$, $c_{1}$ and $c_{2}$ are arbitrary constants; And the …
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