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Dynamics of the Jacobian matrices arising in three-dimensional Euler equations: application of Riccati theory

An attempt is made to remove, at least partially, the non-local nature associated with the pressure term of the incompressible Euler equations. We consider the dynamics of the Jacobian matrix J(t) relating spatial and material coordinates in incompressible three-dimensional Euler equations. By applying the theory of matrix-valued Riccati equations to the velocity gradient tensor V(t) assuming time-symmetric pressure Hessian, we derive an identity which is local in material coordinates for a fluid particle with detV(0)≠0, as a result of invariance under time-reversal. This imposes constraints on the time evolution of the velocity gradients and generates a chain of exact infinite relationships on the Taylor coefficients of V(t) in time. Some of the first few are explicitly given. As a corollary, we prove that if the evolution of the velocity gradient is symmetric with regard to time V(t)=V(−t), it must be a constant and . The key results herein may be summarized in equations (3.10) and (3.11).