Abstract
We study the unique continuation properties of solutions of the Navier-Stokes equations. We take advantage of rotation transformation of the Navier-Stokes equations to prove the “logarithmic convexity” of certain quantities, which measure the suitable Gaussian decay at infinity to obtain the Gaussian decay weighted estimates, as well as Carleman inequality. As a consequence we obtain sufficient conditions on the behavior of the solution at two different timest0=0andt1=1which guarantee the “global” unique continuation of solutions for the Navier-Stokes equations.
Highlights
In this paper, we study the unique continuity of the NavierStokes equations: ut − Δu + u ⋅ ∇u + ∇p = 0 in Rn × (0, 1),∇ ⋅ u = 0 in Rn × (0, 1) . (1)For (1), the existence of the Leray solutions [1] can be found in [2,3,4]
In order to simplify the equations, we introduce q, which is the curl of the solutions of (1): qij
Combining with (64), it implies (49). This completes the proof of Lemma 4
Summary
In [18], Escauriaza et al proved a backward uniqueness result for the heat operator with variable lower order terms, which implies the full regularity of L3,∞-solutions of the three-dimensional NavierStokes equations. In [23], Saut and Scheurer proved a unique continuation theorem when L was a second order parabolic equation in the first section. Their proof is simple and based on the derivation of a Carleman estimate which is reminiscent of the classical Carleman estimates for second order elliptic operators. Our aim is to prove the the following unique continuation theorem of Navier-Stokes equations (1).
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