Towards a sufficient criterion for collapse in 3D Euler equations

Abstract

A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial conditions. Blow-up of this quantity means that solution of the Hopf equation in 3D can not be continued in the Sobolev space H 2 (R 3 ) for infinite time.

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory in Medium

“…In this case the expression (16) exactly coincides with the formula (30) given in [45] for the Lagrangian time evolution of the matrix of the first derivatives of the velocity which must satisfy the three-dimensional Hopf equation (10) (when (10)). In particular, in the one-dimensional case when n = 1, in the Lagrangian representation from (11) and (13), we obtain a particular case of the formula (16): (17) where a is the coordinate of a fluid particle at the initial time t ¼ 0.…”

“…In particular, in the one-dimensional case when n = 1, in the Lagrangian representation from (11) and (13), we obtain a particular case of the formula (16): (17) where a is the coordinate of a fluid particle at the initial time t ¼ 0. The solution (17) also coincides with the formula (14) in [45] and describes the catastrophic process of collapse of a simple wave in a finite time t 0 whose estimate is given above on the basis of the solution to Eq. (13) in the case n ¼ 1 with the use of the Euler variables.…”

“…The second example, which is represented here, also gives new perspectives on the basis of the new exact solution (in the Euler variables) for n-dimensional Hopf equation because this equation is known as the possible model for weak nonlinear optic problems [46]. The importance of the new solution is connected with its Euler form in dependence from space variables, which are not represented in the solution of the Burgers-Hopf equation well known before (see [45] and others).…”

“…The condition of existence of a real positive solution of Eq. (13) (e.g., see (14)) is the necessary and sufficient condition of implementation of the singularity (collapse) of the solution, as distinct from the sufficient but not necessary integral criterion which was proposed in [45] (see formula (38) in [45]) and has the form: (15) In fact, in accordance with this criterion proposed in [45], the collapse of the solution is not possible in the case of the initial divergence-free velocity field, i.e., when div u 0 ! ¼ 0.…”

“…The identities (45) and (46) From (44), in the three-dimensional case, we get on the basis of (47) that all three components of the vector B m 0. For each m ¼ 1, 2, 3, we get identical zeroing separately for the sum of terms proportional to t and separately for the sum of the terms proportional to t 2 .…”