On the uniqueness of compressible fluid motions
On the uniqueness of compressible fluid motions
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DOI：
10.1007/bf00284180
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发表时间：
1959
影响因子：
2.5
通讯作者：
J. Serrin
中科院分区：
文献类型：
--
作者：
J. Serrin
关于任何应用数学问题，必须回答的基本问题之一是它是否集好，也就是说，解是否确实存在，以及它们是否唯一。这里我们将讨论可压缩流体流动的初值问题，我们将特别研究其解的唯一性。在w3（粘性流体）和w4（非粘性流体）中给出了问题的精确表述。粗略地说，这些定理断言有界区域~ v'=~(t)内的流体运动是唯一由其初始速度、温度和密度分布以及某些边界条件决定的。边界条件的形式对于非粘性，非导热流体特别有趣，事实上，当流体以相对超音速离开r时，边界条件在任何一点上都是多余的。这一结果与流动区域中特征流形的存在密切相关，并使我们能够证明一维运动中COURANT & FRIEDRICHS给出的空间流动类型定理。不可压缩黏性流体初值问题的唯一性，早在1929年就由E. Fok证明了。他的证明同样适用于非粘性情况，被DOLIDZE重新发现并更优雅地呈现出来（参见参考文献[renee[15]]的72页，以获得这项工作的概述）。最近，GgAFFI表明，当压力和密度满足压电关系~----I(0)与if (Q)>时，可压缩流体的初值问题是唯一的。我们的工作是对GRAFFI的工作的扩展，因为我们没有对流体的状态方程做任何假设，除了要求它满足某些普遍的热力学条件。从物理和数学的角度来看，这种推广是非平凡的，因为众所周知，压力和密度之间的关系通常与三变量流体的运动不相容，无论它是否具有粘性。此外，对一般情况的处理要求我们使用控制流体运动的全套方程，包括完整的能量方程。该证明使用了经典能量法，该方法最初由HADAMARD， ZAREMBA和FRIEDRICHS & LEWY在与纯数学问题有关时开发，并由REYNOLDS， ORR和Fok用于研究不可压缩流体运动。与这些方法相比，我们的处理方法最新颖的特点也许是系统地利用输运方程来推导能量积分恒等式，从而以一种自然的方式考虑到流动方程中对流项的特殊形式。
One qf the fundamental questions which should be answered concerning any problem of applied mathematics is whether it is well set, that is, whether solutions actually exist and whether they are unique. We shall be concerned here with the initial value problem fo'r compressible-fluid flow, and we shall study in particular the uniqueness of its solutions. Aft exact statement of the problem will be found in wt, while the main theorems are given in w 3 (viscous fluids) and w 4 (non-viscous fluids). Stated roughly, these theorems assert that fluid motion in a bounded region~ v'=~(t) is uniquely determined by its initial velocity, temperature, and density distribution together with certain boundary conditions. The form of the boundary conditions is especially interesting for non-viscous, non-heat conducting fluids, it being fourid, in fact, that such conditions are superfluous at any point where the fluid is leaving r at a relatively supersonic speed. This result is intimately associated with the existence of characteristic manifolds in the flow region,, and allows us to prove for spatial flows a theorem of the type given by COURANT & FRIEDRICHS for one-dimensional motions.The uniqueness of the initial value problem for incompressible viscous fluids was proved as long ago as~ 929 by E. Fok. His proof, which applies equally well to the non-viscous case, was rediscovered and presented more elegantly by DOLIDZE (see w 72 of refe~ renee [15] for a resum~ of this work). More recently, GgAFFI showed that the initial value problem for compressible fluids is unique, provided that the pressure and density satisfy a piezotropic relation~----I (0) with if (Q)> 0. Our work is an extension of GRAFFI'S, in that we make no assumption'concerfling the equation of state of the fluid, beyond requiring it to satisfy certain universal thermodynamic conditions. This generalization is non-trivial from both the physical and/-nathematical point of view, for it is well known that a relation between pressure and density is usually incompatible with the motion of a tri-variate fluid, whether or not it is viscous. Moreover, treatment of the general case requires that we use the full set of equations governing the fluid motion, including the complete energy equation. The proof makes use of the classical energy method, developed originally by HADAMARD, ZAREMBA, and FRIEDRICHS & LEWY in connection with purely mathematical problems, and by REYNOLDS, ORR and Fok for the study of incompressible fluid motions. Perhaps the most novel feature of our treatment, in comparison with these, is th~ systematic utilization of the transport equation for the derivation of energy integral identities, thus taking into account in a natural way the special form of the convection terms in the flow equations.