科研之友
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摘要
We consider the three-dimensional Navier-Stokes equations on the whole space R-3 and on the three-dimensional torus T-3. We give a simple proof of the local existence of (finite energy) solutions in L-3 for initial data u(0) is an element of L-2 boolean AND L-3, based on energy estimates and regularisation of the initial data with the heat semigroup. We also provide a lower bound on the existence time of a strong solution in terms of the solution v(t) of the heat equation with such initial data: there is an absolute constant epsilon > 0 such that solutions remain regular on [0, T] if parallel to u(0)parallel to(3)(L3) integral(T)(0) integral(R3) vertical bar del v(S)vertical bar(2)vertical bar v(S)vertical bar dx dt <= epsilon. This implies the u is an element of C-0 ([0, T]; L-3) regularity criterion due to von Wahl. We also derive simple a priori estimates in L-p for p > 3 that recover the well known lower bound parallel to u(T-t)parallel to(Lp) >= ct(-(p-3)/2p) on any solution that blows up in L-P at time T. The key ingredients are a calculus inequality parallel to u parallel to(p)(L3p) <= c integral vertical bar u vertical bar(p-2)vertical bar del u vertical bar(2)(valid on R-3 and for functions on bounded domains with zero average) and the bound on the pressure parallel to p parallel to(Lr) <= c(r)parallel to u parallel to(2)(L2r), valid both on the whole space and for periodic boundary conditions.
- 出版日期2014
