Local Estimates for Regular Solutions

Abstract

The aim of this chapter is to show the estimate

$$\displaystyle \begin{array}{@{}rcl@{}} \|v\|{ }_{W^{2,1}_2(\varOmega ^t)} \le M, \end{array} $$

where M depends on some norms of data but does not depend on time t ∈ (0, T] explicitly. The dependence on time is only through integrals with respect to time of data functions f, d and their time and space derivatives. To prove the above inequality, we need smallness of the following quantity

$$\displaystyle \begin{array}{@{}rcl@{}} \varLambda _2(t) = \int _0^t (\|d_{x'}\|{ }_{H^1(S_2)}^2+\|d_t\|{ }_{H^1(S_2)}^2+ |f_3|{ }_{L_{4/3}(S_2)}^2+|g|{ }_{L_{6/5}(\varOmega )}^2)\, dt' \\ +\mathcal {A}\sup _t\|d_{x'}\|{ }_{W^1_{3/2}(S_2)}^2 + |h(0)|{ }^2_{L_2(\varOmega )}, \end{array} $$

where $h=v_{,x_3}, g=f_{,x_3}$, and $\mathcal {A}$ estimates the weak solution (see Chap. 3). To achieve this, we consider the problems for variables $h, q=p_{,x_3}$ and $\chi = v_{2,x_1}-v_{1,x_2}.$ Then for sufficiently small Λ ₂(t) we can apply some fixed point argument and use energy estimates for solutions to problems on h, q and χ to obtain the desired bound on v by M.