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We give a new sufficient criteria to prove the uniqueness of theincompressible Euler equation in dimension $N\geq2$. In their celebrated worksby V. Scheffer [18], A. Shnirelman [19], C. De Lellis and L. Sz\'ekelyhidi Jr.[7] they have obtained the nonuniqeness of weak solutions of incompressibleEuler equation. Here we obtain uniqueness criteria for the same equation undersome mild regularity condition on weak solutions. Our proof is simple and canbe employed to other equations like inhomogeneous incompressible Euler andEuler-Boussinesq equations. One of the key ingredients in our proof iscommutator estimate [5, 11].

A sufficient condition for uniqueness of weak solutions of the incompressible Euler system