一类修正的Navier-Stokes方程的广义解的正则性

Regularity of Generalized Solutions for a Class of Modified Navier-Stokes Equations

1968年提出以形如v_+v_kv_(ik)-(T_(ik)(v_(jl))_=-q_+f_i(i=1,2,3,v_(ik)=v_+v_的所谓修正的Navier-Stokes方程来代替经典的Navier-Stokes方程.证明了它和连续性方程联立的初边值问题的广义解的整体的存在与唯一性.本文证明了上述广义解有如下正则性质:1~*若初值a∈(Ω)∩W_2~1(Ω),外力,f∈L^1(0,T_1,W_2~1(Ω)),并且或者T>0充分小或者‖a‖充分小,则存在v∈L~∞(0,T;W_2~1(Ω)),v∈L_2(Q_T);2~*若a(Ω)∩W_2~1(Ω),f,f_s∈L^1(0,T;L_2(Ω)),‖v(t)‖<∞,并且假定T_(ik)几乎处处有可测的导数Tih/v_(fl)|T_(ik)/v_(il)|≤y(||),式中γ(r)是r∈的非负连续函数,,则存在v_t∈L~∞(0,T;L_2(Ω)),v_t∈L_2(Ω_t).

In 1968,O.A.Ladyzhenskaya suggested to substitute the following modified Navier-Stokes equations v_u+v_kv_(ik)-(T_u(v_(jl)))_(zk)=-q_(zi)+f_i.(i=1,2,3) (v_(ik)=v_(izk)+v_(hxi))(*) for the classical Navier-Stokes equations.She proved a global existence and uniquencess theorem for the generalized solution of the initial-boundary value problem for the combined system of (*) together with the continuity equation (see Zap.Nauch.Sem.Leningrad,Otdel.Mat.Inst.Steklov,Tom 7(1968),126— 154,in Russian;see also Appendix in Math.problems in the Dynamics of Viscous Incompressible Flow, Moscow;Nauka,1970,Second Russian ed.).In this paper,we show that the generalized solution mentioned above has the following regularity properties;1° If initial value a∈J(Ω)∩W_2~1(Ω),external force f∈L^1 (O,T;W_2~1(Ω)),and if T>0 is sufficiently small or ‖_x‖ is sufficiently small,then ∈L~∞(O,T;W_2~1 (Ω))and ∈L_2(Q_T);2° If a∈(Ω)∩W_2~3(Ω),f,f_l∈L^1(O,T;L_2(Ω)),and also ‖_zv(t)‖<∞, and if the derivatives T_u/v_(jl)of the nonlinear items T_(ik)are measurble and their absolute values are bounded by nonnegative continuous function γ(||)(||=,then v_t∈L~∞(O,T;L_2(Ω)) and _zv_t∈L_2(Q_T).