Weak Solutions with Decreasing Energy of Incompressible Euler Equations

摘要

The physical meaning of weak solutions of the Euler equations is not quite clear.211u001eThe authors may assume as a hypothesis, that the velocity field of a slightly 211u001eviscous and slightly compressible fluid tends to a weak solution of the Euler 211u001eequations, when both viscosity and compresibility tend to zero (the authors admit 211u001ean arbitrary rheology of the fluid, whatever nonlinear and nonlocal it is; 211u001eimportant is only the fact that the viscosity is nonzero). As the authors know 211u001efrom the experiments, the kinetic energy of the turbulent flow (in the absence of 211u001eexternal forces) decreases, for the fluid develops large velocity gradients, 211u001edissipating considerable amount of energy. It is a well estabished experimental 211u001efact, that the rate of the energy decay does not depend on the viscosity, if the 211u001elast is small enough. Thus, the 'true' weak solution, describing a turbulent 211u001eflow, should have decreasing kinetic energy. The first natural question is, 211u001ewhether there exist weak solutions with monotonically decreasing kinetic energy.