COUNTEREXAMPLES OF COMMUTATOR ESTIMATES IN BESOV AND THE TRIEBEL-LIZORKIN SPACES RELATED THE EULER EQUATIONS.

This paper deals with the Kato Ponce-type commutator estimates in the Besov space Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and the Triebel-Lizorkin space Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. related to the Euler equations describing the motion of perfect incompressible fluids. We investigate the relation between the optimal bound of the commutator estimates and the solvability of the Euler equations. In particular, we show that these commutator estimates fail in Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. with the critical differential order s = n/p + 1 and various exponents p and q. [ABSTRACT FROM AUTHOR]