Abstract. It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.
We consider the regularity criteria for the incompressible Navier-Stokes equations connected with one velocity component. Based on the method from [4] we prove that the weak solution is regular, provided u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 3 4 + 1 2s , s > 10 3 or provided ∇u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 19 12 + 1 2s if s ∈ (30 19 , 3] or 2 t + 3 s ≤ 3 2 + 3 4s if s ∈ (3, ∞]. As a corollary, we also improve the regularity criteria expressed by the regularity of ∂p ∂x 3 or ∂u 3 ∂x 3 .
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