Semi-strong Solutions, m = 2, k = 1

Abstract

In this chapter we prove Theorem 1.4.3 on the existence and uniqueness of semi-strong solutions of problem (VKH) when m = 2 (recall that, by Definition 1.4.1, if m = 2 there is only one kind of semi-strong solution, corresponding to k = 1). Accordingly, we assume that

$$\displaystyle{ u_{0} \in H^{3}\,,\qquad u_{ 1} \in H^{1}\,,\qquad \varphi \in S_{ 2,1}(T) = C([0,T];H^{5}) }$$

(4.1)

[recall (1.137)], and look for solutions of problem (VKH) in the space $\mathcal{X}_{2,1}(\tau )$, for some τ ∈ ]0, T].