当密度 $varrho$ 的正则性没有$L^2$ 时, 我们用如下的震荡估计: $$eelabel{eq} sup_{k>1}limsup_{delta o 0^+} sen{T_k(varrho_delta)-T_k(varrho)}_{gamma+1} leq L(Omega,bf,bg,m), eee$$其中 $$ex T_k(t)=left{a{ll} t,&0leq tleq k,\ k,&tgeq k. ea ight. eex$$ 证明: $$ex & &limsup_{delta o 0^+}int sev{T_k(varrho_delta)-T_k(varrho)}^{gamma+1}\ & &leqlimsup_{delta o 0^+}int sex{varrho_delta-varrho}^gamma sex{T_k(varrho_delta)-T_k(varrho)}\ & &leqlimsup_{delta o 0^+}int sex{varrho_delta^gamma-varrho^gamma} sex{T_k(varrho_delta)-T_k(varrho)}\ & &=int overline{varrho^gamma T_k(varrho)}-overline{varrho^gamma}T_k(varrho)-varrho^gamma overline{T_k(varrho)}+varrho^gamma T_k(varrho)\ & &=int overline{varrho^gamma T_k(varrho)} -overline{varrho^gamma}overline{T_k(varrho)}\ & &quad +int sex{overline{varrho^gamma}-varrho^gamma}sex{overline{T_k(varrho)}-T_k(varrho)}\ & &leq int overline{varrho^gamma T_k(varrho)} -overline{varrho^gamma}overline{T_k(varrho)}quadsex{mbox{凸性}}\ & &=(lambda+2mu)int overline{T_k(varrho)Divbu}-overline{T_k(varrho)}Divbuquadsex{mbox{有效粘性通量}}\ & &=(lambda+2mu)limsup_{delta o 0^+} int T_k(varrho_delta)Divbu_delta -overline{T_k(varrho)}Divbu_delta\ & &leq (lambda+2mu) limsup_{delta o 0^+} sez{sen{bu_delta}_2cdotsen{T_k(varrho_delta)-overline{T_k(varrho)}}_2}\ & &leq Climsup_{delta o 0^+} sen{T_k(varrho_delta)-T_k(varrho)}_{gamma+1}. eex$$