Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics
Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics
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DOI：
10.1090/s0002-9939-2012-11591-6
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发表时间：
2012-01
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通讯作者：
R. Danchin
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作者：
R. Danchin
我们给出了无粘Boussinesq系统的解决方案的寿命的下限。在二维中，我们指出当初始（相对）温度趋于零时，它趋于无穷大。据我们所知，这是无粘Boussinesq系统的第一个此类结果。顺便，我们在N维情形下给出了（独立的）连续性准则.在本文的第二部分中，我们的方法适用于处理轴对称不可压欧拉方程与漩涡。介绍一个完美的均匀不可压缩流体的速度u = u（t，x）和压力P = P（t，x）场的演化由以下欧拉方程控制：（0. 1）{tu+ u ·u+ P = 0，div u = 0.关于欧拉方程的适定性问题有大量的文献。粗略地说，它们可以在嵌入有界Lipschitz函数集合C 0，1中的任何合理的Banach空间中在时间上局部求解（参见例如[1，4，6，12，13，17，19，22]）。在二维情况下，众所周知，欧拉方程是全局适定的充分光滑的初始数据。这一值得注意的事实依赖于涡量ω：=沿着速度场流动的守恒，并且在W. Wolibner [20]和V. Yudovich [21]。然而，在更多的物理相关的情况下，这种守恒性质不再成立，例如（1）（0.1）的三维设置，（2）非均匀不可压缩的理想流体，（3）受到浮力的无粘性流体，该浮力由速度流体平流（下面所谓的无粘性Boussinesq系统）。因此，一般（甚至光滑或小）数据的全局存在性问题对于上述三种情况仍然是开放的。在最近的工作[9]中，已经表明，对于稍微非均匀的二维不可压缩流体，当非均匀性趋于零时，寿命趋于无穷大。本文主要研究第一项和第三项的寿命。更准确地说，在本文的第一部分，我们将考虑无粘Boussinesq系统：（0.2）θ = u ·θ = 0，θ = u·u+ p = θeN，div u = 0。这里，相对温度θ = θ（t，x）是一个真实的值函数，eN代表单位垂直矢量。2010年数学学科分类。35Q35，76B03 1它不需要是非负的，因为它表示与某个参考温度的差异。1
We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the N -dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl. Introduction The evolution of the velocity u = u(t, x) and pressure P = P (t, x) fields of a perfect homogeneous incompressible fluid is governed by the following Euler equations: (0.1) { ∂tu+ u · ∇u+∇P = 0, div u = 0. There is a huge literature concerning the well-posedness issue for Euler equations. Roughly, they may be solved locally in time in any reasonable Banach space embedded in the set C0,1 of bounded Lipschitz functions (see e.g. [1, 4, 6, 12, 13, 17, 19, 22]). In the two-dimensional case, it is well known that Euler equations are globally well-posed for sufficiently smooth initial data. This noticeable fact relies on the conservation of the vorticity ω := ∂1u 2 − ∂2u 1 along the flow of the velocity field, and has been first proved rigorously in the pioneering works by W. Wolibner [20] and V. Yudovich [21]. This conservation property is no longer true, however, in more physically relevant contexts such as (1) the three-dimensional setting for (0.1), (2) nonhomogeneous incompressible perfect fluids, (3) inviscid fluids subjected to a buoyancy force which is advected by the velocity fluid (the so-called inviscid Boussinesq system below). As a consequence, the problem of global existence for general (even smooth or small) data is still open for the above three cases. In a recent work [9], it has been shown that for slightly nonhomogeneous two-dimensional incompressible fluids, the lifespan tends to infinity when the nonhomogeneity tends to zero. The present paper is mainly dedicated to the study of the lifespan for the first and third item. More precisely, in the first section of the paper, we shall consider the inviscid Boussinesq system: (0.2) ∂tθ + u · ∇θ = 0, ∂tu+ u · ∇u+∇P = θeN , div u = 0. Here the relative temperature θ = θ(t, x) is a real valued function and eN stands for the unit vertical vector. 2010 Mathematics Subject Classification. 35Q35,76B03. 1It need not be nonnegative as it designates the discrepancy to some reference temperature. 1