Non-stationary flow of an ideal incompressible liquid

• Published in: U.S.S.R. computational mathematics and mathematical physics 1963, Vol.3 (6), p.1407-1456
• Main Author: Yudovich, V.I.
• Format: Article
• Language: English
• ISSN: 0041-5553
• DOI: 10.1016/0041-5553(63)90247-7

Our aim in this paper is to prove that there is a single-valued solution “in the whole” of the two-dimensional problem with initial data for the equations of motion of an ideal incompressible liquid contained in some vessel. The existence and uniqueness of a solution on a sufficiently small time interval has been proved for sufficiently smooth data in the three-dimensional problem by Gyunter (see [1]) and Lichtenstein (see [2], 482–493). Here, we consider generalized solutions and give detailed proofs of the results given in [3]. A separate paper will be devoted to smooth solutions and to the problem of the flow of a liquid through a given region (see [4]). Let us give a brief summary of the contents of the paper. Section 1 is devoted to a priori estimates of the solutions of the Euler equations. The basic result is an a priori estimate of the maximumof the curl for a certain class of flow patterns which includes plane and axisymmetric cases. We do not exclude the case when the vessel containing the liquid is deformed with time in a given way. In Section 2 we give a brief summary of certain properties of elliptic equations and integrals of potential type (see [3]. [5]–[7]) which will be needed later. We also introduce certain space functionals and study their properties. In Sections 3–4 we give the definition of a generalized solution and prove its uniqueness and existence for a simply connected flow region. Certain properties of the generalized solution, in particular its smoothness, are examined. A similar examination is made in Section 5 for multiply connected regions. It is shown in Section 6 that the resulting generalized solution enables us to determine the pressure and the trajectory of the particles of liquid, and the results of an examination of the external problem are given.