On 2D Rayleigh-Taylor instabilities
On 2D Rayleigh-Taylor instabilities
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二维瑞利-泰勒不稳定性
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
V. Kamotski
中科院分区：
文献类型：
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作者：
Dérivées Partielles;G. Lebeau;V. Kamotski
1引言考虑在平面上的流动的两个理想的不可压缩流体的恒定密度ρ ± > 0的重力场，与ρ + = ρ −。速度场满足欧拉方程<$ρu <$t + u ·<$ρu = − <$p + ρg（1）div u = 0（2）ρ t + div（ρu）= 0（3），初始数据u（x，0）= u 0（x）。(4)我们假设u 0（x）满足连续性方程（2），并且涡度ω 0（x）= rot u 0的假设集中在分离两种流体的某条曲线上，即我们假设rot 0将平面分成两个区域，其中流体分别具有恒定的密度ρ +和ρ-。所以我们有ω 0 =<$δ <$0（5），其中<$= [u||]是速度的正切到θ 0分量的跳跃。将rot算子应用于（1），我们发现涡量ω =沿着每个子域的流动形式上是恒定的，其中ρ是恒定的：ω t ω + u ·ω t = 0（6）因此，对于t > 0，人们期望涡量保持集中在某条曲线ω t上，一条与时间相关的曲线ω t将两种流体分开。这个问题被称为瑞利-泰勒不稳定性。在[SS 85]中已经表明，在周期性界面接近平坦线的情况下，并且具有解析数据，0，这个问题在时间上是局部适定的。另一方面，在[Leb 02]和[Wu]中，我们证明了对于Kelvin-Helmholtz问题，即ρ + = ρ −，g = 0的情形，涡面的演化问题是强不适定的，在这个意义上，如果α不为零，则数据的解析性是得到具有C1 +α界面α > 0的局部时间解的必要条件。本文的主要目的是将上述关于Kelvin-Helmholtz不稳定性的结果推广到更复杂的Rayleigh-Taylor情形。为了固定几何形状，我们假设界面是平面上的一条封闭的简单曲线（周期曲线的情况）。
1 Introduction Consider in the plane the flow of two ideal incompressible fluids of constant densities ρ ± > 0 in the gravity field, with ρ + = ρ −. Velocity field satisfies the Euler equation ∂ρu ∂t + u · ∇ρu = −∇p + ρg (1) div u = 0 (2) ρ t + div (ρu) = 0 (3) with initial data u(x, 0) = u 0 (x). (4) We suppose, that u 0 (x) satisfies the continuity equation (2), and the assumption that vorticity ω 0 (x) = rot u 0 is concentrated on some curve Σ 0 separating the two fluids, i.e. we suppose Σ 0 splits the plane into two domains Ω 0 + and Ω 0 − , in which the fluids have constant densities ρ + and ρ − respectively. So we have ω 0 = Ωδ Σ 0 (5) with Ω = [u || ] being the jump of the tangent to Σ 0 component of velocity. Applying the rot operator to (1) we find that vorticity ω = ∂ x u y − ∂ y u x is formally constant along the flow in each subdomain where ρ is constant : ∂ t ω + u · ∇ω = 0 (6) Thus for t > 0 one expects the vorticity to remain concentrated on some curve Σ t , with a time dependant curve Σ t separating the two fluids. This problem is known as the Rayleigh-Taylor instability. It has been shown in [SS85] that in the case of a periodic interface close to a flat line and with analytic data Ω, Σ 0 , this problem is locally in time well posed. On the other hand, in [Leb02] and [Wu], it has been proved that for the Kelvin-Helmholtz problem, i.e in the case ρ + = ρ − , g = 0 the evolution problem of the vortex sheet is strongly ill-posed in the sense that if Ω does not vanish, analyticity of the data is a necessary condition to get a local in time solution with a C 1+α interface Σ with α > 0. The main goal of this paper is to extend the above results on Kelvin-Helmholtz instability to the more involved Rayleigh-Taylor case. In order to fix geometry, we suppose that the interface Σ t is a closed simple curve in the plane (the case of periodic …