Approximating the Navier-Stokes equations on R^3 with large periodic domains
报告题目(Title)：Approximating the Navier-Stokes equations on R^3 with large periodic domains
报告人(Speaker)：James Robinson (University of Warwick)
地点(Place)：Zoom ID: 620 083 67755 密码: 850302
时间(Time)：11月13日（周五） 18:00—19:00 (北京时间)
邀请人(Inviter)：Calvin Khor, 吴家宏，许孝精，袁迪凡
报告摘要
I will consider solutions u_α of the three-dimensional Navier–Stokes equations on the periodic domains Q_α := (−α, α)^3 as the domain size α → ∞, and compare them to solutions of the same equations on the whole space. For compactly-supported initial data u_α^0 ∈ H^1 (Q_α), an appropriate extension of u_α converges to a solution u of the equations on R^3, strongly in L^r(0, T; H^1(R^3)), r ∈ [1, 4) (the result is in fact more general than this). The same also holds when u_α^0 is the velocity corresponding to a fixed, compactly-supported vorticity. Such convergence is sufficient to show that if an initial compactly-supported velocity u_0 ∈ H^1(R^3) or an initial compactly-supported vorticity ω_0 ∈ H^1(R^3) gives rise to a smooth solution on [0, T∗] for the equations posed on R^3, a smooth solution will also exist on [0, T∗] for the same initial data for the periodic problem posed on Q^α for α sufficiently large; this illustrates a ‘transfer of regularity’ from the whole space to the periodic case.
