Landslides - Causes, Types and Effects.pdf

302 Jiří Nedomawith approximate initial conditions〈u m (x, 0) − u 0 (x), v j 〉 Ω =0, 〈u ′ m(x, 0) − u 1 (x), v j 〉 Ω =0. (99)The system represents an initial value problem for a system of ordinary differentialequations of the second order for a.e. t ∈ I. The solvability of such a system follows fromthe theory of systems of ordinary differential equations. The global existence follows fromthe a priori estimates.A priori estimates:Firstly the term∫∫ t0∪ k,l Γ kl c1δ F klc ([u ′ mt(τ)] kl )[u ′ mn(τ)] kl+ ·∇ϕ ε ([u ′ mt(τ)] kl )[v jt ] kl dsdτ ≥ 0.In order to pass to the limit m →∞of the Faedo-Galerkin parameter we need suitablea priori estimates.Multiplying (98) by c ′ mj(t), summing over j, integrating over [0,t], t ∈ I, and puttingthen we havem u ′ m (t) − u′ 2 (t) = ∑c ′ mj (t)v j,j=1∫ t{(u ′′ m(τ), u ′ m(τ)) + (αu ′ m(τ), u ′ m(τ)) + a (0) (u m (τ), u ′ m(τ)) +0〈 〉1+a (1) (u ′ m(τ), u ′ m(τ)) +δ [u′ mn(τ)] kl+, [u ′ mn(τ)] kl +∪ k,l Γ kl c∫1+δ F c kl ([u ′ mt(τ)] kl )[u ′ mn(τ)] kl+ ·∇ϕ ε ([u ′ mt(τ)] kl )[u ′ mt(τ)] kl ds}dτ ==∪ k,l Γ kl c∫ t0{(f(τ), u ′ m(τ) − u ′ 2(τ)) + (u ′′ m(τ),u ′ 2 (τ)) + a(0) (u m (τ), u ′ 2 (τ)) + a(1) (u ′ m (τ), u′ 2 (τ))}dτ. (100)We estimate separate terms in (100). In respect of the definition of the space Vwe have [u 2n ] kl = 0 on ∪ k,l Γ klc , and furthermore, we have u ′ m(0) = u 1 and as∇ϕ ε ([u ′ mt] kl )[u ′ mt] kl ≥ 0 then the last term on the left-hand side is ≥ 0. Completingall estimates, then after some modification, using the Korn inequality and the Gronwalllemma, there exists a constant C, which does not depend on the Faedo-Galerkin index mand parameters δ and ε, such that we have‖u ′ m(t)‖ 2 L ∞ (I;L 2,N (Ω)) + ‖u m(t)‖ 2 L ∞ (I;H 1,N (Ω)) + ‖u′ m(t)‖ 2 0,1,Q+ 1 δ ‖[u′k mn(t) − u ′l mn(t)] + ‖ 2 0,Γ c(t) ≤ C, (101)