Regularity of Leray-hopf Solutions to Navier-stokes Equations (i)-critical Regularity in Weak Spaces
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چکیده
We consider the regularity of Leray-Hopf solutions to impressible Navier-Stokes equations on critical case u ∈ L 2 w (0, T ; L ∞ (R 3)). By a new embedding inequality in Lorentz space we prove that if u L 2 w (0,T ;L ∞ (R 3)) is small then as a Leray-Hopf solution u is regular. Particularly, an open problem proposed in [8] is solved.
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Regularity of Leray-hopf Solutions to Navier-stokes Equations (i)-critical Interior Regularity in Weak Spaces
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Theorem 1.1. Suppose u is a Leray-Hopf solution to the Navier-Stokes equation (1.1) with initial data u0 ∈ L(R) and blows up as t → T . Then (1) (T − t) 14‖∇xu(t)‖L2(R3) → 0, as t → T ; (2) (T − t) 1 2‖u(t)‖L∞(R3) → 0, as t → T. Here u : (x, t) ∈ R × (0, T ) → R is called a weak solution of (1.1) if it is a Leray-Hopf solution. Precisely, it satisfies (1) u ∈ L(0, T ;L(R)) ∩ L(0, T ;H(R)), (2) ...
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