2412 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS

In addition to this, t→ u(t,ω) is continuous into H and satisfies

u(·,ω) ∈ L∞ ([0,T ] ;H)∩L2 ([0,T ] ;W ) .

If, in addition to the above, u0 ∈ L2 (Ω;H) and f ∈ L2 ([0,T ]×Ω;W ′) and

q ∈ L2 ([0,T ]×Ω;V ) ,

then the solution u is in L2 ([0,T ]×Ω;H)∩L2 ([0,T ]×Ω;W ).

Proof: The last claim follows from the estimates used in the Galerkin method, takingexpectations and passing to a limit. To verify the continuity into H, one can observe thatfrom the integral equation, u is continuous into V ′. One has

|u(t)|2H =∞

∑k=1

(u(t) ,wk)2 =

∞

∑k=1⟨u(t) ,wk⟩2 ,

and so t→ |u(t)|2H is lower semi-continuous. Since it is in L∞, this implies this function isbounded. Hence the continuity into V ′ and density of V in H implies that u(t) is weaklycontinuous into H. Then one can use the formulation in Theorem 70.4.5 to verify t →|u(t)|H is continuous and apply uniform convexity of the Hilbert space H.

One can replace q(t,ω) with q(t,ω,u) and f(t,ω) with f(t,ω,u) in the above with nochange in the argument, provided it is assumed that

(t,ω,u)→ q(t,ω,u) , f(t,ω,u)

are product measurable, continuous in (t,u) and bounded.

70.5 A Friction contact problemIn this section we will consider a friction contact problem which has a coefficient of frictionwhich is dependent on the slip speed.

üi = σ i j, j(u, u̇)+ fi for (t,x) ∈ (0,T )×U, (70.5.15)

u(0,x) = u0(x), (70.5.16)

u̇(0,x) = v0(x), (70.5.17)

where U is a bounded open subset of R3 having Lipschitz boundary, along with someboundary conditions which pertain to a part of the boundary of U , ΓC. For x ∈ ΓC,

σn =−p((un−g)+)Cn, (70.5.18)

This is the normal compliance boundary condition.

|σT | ≤ F((un−g)+)µ(∣∣u̇T − U̇T

∣∣) , (70.5.19)

|σT |< F((un−g)+)µ(∣∣u̇T − U̇T

∣∣) implies u̇T − U̇T = 0, (70.5.20)