70.5. A FRICTION CONTACT PROBLEM 2415

Theorem 70.5.2 Let W,U, and Y be as in Theorem 70.5.1 and let

S = {u : ||u(t)||W + ||u′||Lq(0,T ;Y ) ≤ R for t ∈ [0,T ]}

for q > 1. Then S is pre compact in C(0,T ;U).

Now we give an abstract formulation of the problem described roughly in 70.5.15 -70.5.21. We begin by defining several operators. Let M,A : V →V ′ be given by

⟨Mu,v⟩=∫

UCi jkluk,lvi, jdx, (70.5.29)

⟨Au,v⟩=∫

UAi jkluk,lvi, jdx. (70.5.30)

Also let the operator v→ P(u) map V to V ′ be given by

⟨P(u),w⟩=∫ T

0

∫ΓC

p((un−g)+)wndαdt, (70.5.31)

where

u(t) = u0 +∫ t

0v(s)ds (70.5.32)

for u0 ∈Vq. (Technically, P depends on u0 but we suppress this in favor of simpler notation). Let

γ∗T : L2

(0,T ;L2 (ΓC)

3)→ V ′

is defined as

⟨γ∗T ξ ,w⟩ ≡∫ T

0

∫ΓC

ξ ·wT dαdt.

Now the abstract form of the problem, denoted by P , is the following.

v′+Mv+Au+Pu+ γ∗T ξ = f in V ′, (70.5.33)

v(0) = v0 ∈ H, (70.5.34)

where

u(t) = u0 +∫ t

0v(s)ds, u0 ∈Vp, (70.5.35)

and for all w ∈V ,

⟨γ∗T ξ ,w⟩ ≤∫ T

0

∫ΓC

F((un−g)+)µ(∣∣vT − U̇T

∣∣) ·[∣∣vT − U̇T +wT

∣∣− ∣∣vT − U̇T∣∣]dαdt. (70.5.36)

Also f ∈ L2(0,T ;V ′) so f can include the body force as well as traction forces on variousparts of ∂U. If v solves the above abstract problem, then u can be considered a weak