Generalizing the Bardos-LeRoux-Nédélec boundary condition for ...

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The problem BLN condition & Alternatives Effective BC graph Definition Uniqueness Definition Bis Existence & Convergence of Approximations ...Uniqueness, comparison, L 1 contraction. For the proof, by the Kruzhkov’s doubling of variables argument applied “inside Ω” one deduces the “local Kato inequality” (u−û) + (t)ξ ≤ (u0−û0) + t ξ(0, ·) + q + (u, û) · ∇ξ Ω Ω 0 Ω for all ξ ∈ D([0, t] × Ω).
The problem BLN condition & Alternatives Effective BC graph Definition Uniqueness Definition Bis Existence & Convergence of Approximations ...Uniqueness, comparison, L 1 contraction. For the proof, by the Kruzhkov’s doubling of variables argument applied “inside Ω” one deduces the “local Kato inequality” (u−û) + (t)ξ ≤ (u0−û0) + t ξ(0, ·) + q + (u, û) · ∇ξ Ω Ω 0 Ω for all ξ ∈ D([0, t] × Ω). Take for ξ ∈ D(R × RN ) truncation-near-the-boundary functions ξh. We “pay” for this truncation with a new term which is “dissipative”. Indeed, t (∗) as h ↓ 0, 0 q Ω + t (u, û) · ∇ξh −→ − 0 γwq ∂Ω + t (u, û) = − q + (γu, γû) ≤ ??? 0, 0 ∂Ω
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Generalizing the Bardos-LeRoux-Nédélec boundary condition for ...

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