Uniform H^{s-2} Lipschitz bound for the time derivative of a Galerkin solution

(Uniform projected inverse.) Let $M=P_N{A}^0(u_N)P_N$ on $X_N$. Positivity gives $\mathrm{\parallel }{M}^{-1}h{\mathrm{\parallel }}_2\le a^{-1}\mathrm{\parallel }h{\mathrm{\parallel }}_2$. For $v=M^{-1}h$ and $\mathrm{\mid }\alpha \mathrm{\mid }=j\le s-2$, differentiate $P_N(A^{0}v)=h$ and pair with $D^{\alpha }v\in X_N$ to obtain $a\mathrm{\parallel }{D}^{\alpha }v{\mathrm{\parallel }}_2\le \mathrm{\parallel }{D}^{\alpha }h{\mathrm{\parallel }}_2+\mathrm{\parallel }[D^{\alpha },A^0]v{\mathrm{\parallel }}_2\le \mathrm{\parallel }{D}^{\alpha }h{\mathrm{\parallel }}_2+C_R\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{j-1}}$. Induction and summation give $\mathrm{\parallel }{M}^{-1}h{\mathrm{\parallel }}_{H^{s-2}}\le C_R\mathrm{\parallel }h{\mathrm{\parallel }}_{H^{s-2}}$, uniformly in $N$.

(Differentiate the mass equation.) Smoothness of the finite-dimensional ODE permits time differentiation. Set $z=u_N$ and $g={\mathrm{\partial }}_{t}z$. The differentiated Galerkin identity is $M{\mathrm{\partial }}_{t}g=-P_N({\dot{A}}^{0}g+{\dot{A}}^i{\mathrm{\partial }}_{i}z+A^i{\mathrm{\partial }}_{i}g+\dot{B})$, where $\dot{F}=F_t+F_{u}g$ for each coefficient. The compositions $F_t,F_u$ have bounded $H^{s-1}$ norm by [L3], and $g,{\mathrm{\partial }}_{i}z$ have bounded $H^{s-1}$ norm by [L2]. Since $s-1\ge 4$, the algebra bound makes ${\dot{A}}^{0}g$, ${\dot{A}}^i{\mathrm{\partial }}_{i}z$ and $\dot{B}$ bounded in $H^{s-1}$; $A^i{\mathrm{\partial }}_{i}g$ is bounded in $H^{s-2}$ by the multiplier bound. Thus the whole projected source is bounded in $H^{s-2}$ by a constant depending on $R$, without any universal polynomial growth assumption.

(Second derivative and Lipschitz conclusion.) Apply step 1.1 to step 1.2 to obtain $\mathrm{\parallel }{\mathrm{\partial }}_{t}g{\mathrm{\parallel }}_{H^{s-2}}\le C_R$. Integrate on the segment between $t,t^{\mathrm{\prime }}$ using [L5] to get $\mathrm{\parallel }g(t)-g(t^{\mathrm{\prime }}){\mathrm{\parallel }}_{H^{s-2}}\le C_R\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid }$. This proves both claims.