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 Equivalent definition of solution Proposition (Entropy solution) Let u ∈ L ∞ . The assertions (i),(ii) are equivalent : (i) (“def. with traces”) u is an entropy solution in the above sense (ii) (“def. a-la Carrillo” + technicalities ) The function u verifies: ∀k ∈ R ∀ξ ∈ D([0, T ) × Ω) + T ± ± −(u − k) ξt − q (u, k) · ∇ξ − (u0 − k) 0 Ω Ω ± ξ(0, ·) ≤ Ck∧ β(t,x)(k) − ϕν(x)(k) ∓ ξ(t, x). Σ Here, Ck is a constant that depends on u∞ and on k .
The problem BLN condition & Alternatives Effective BC graph Definition Uniqueness Definition Bis Existence & Convergence of Approximations Equivalent definition of solution Proposition (Entropy solution) Let u ∈ L ∞ . The assertions (i),(ii) are equivalent : (i) (“def. with traces”) u is an entropy solution in the above sense (ii) (“def. a-la Carrillo” + technicalities ) The function u verifies: ∀k ∈ R ∀ξ ∈ D([0, T ) × Ω) + T ± ± −(u − k) ξt − q (u, k) · ∇ξ − (u0 − k) 0 Ω Ω ± ξ(0, ·) ≤ Ck∧ β(t,x)(k) − ϕν(x)(k) ∓ ξ(t, x). Σ Here, Ck is a constant that depends on u∞ and on k . And if the sets Σ ± (k) := {(t, x) ∈ Σ | k ∈ K±(t, x)} are “regular enough” then (i),(ii) are also equivalent to (ii’) (“def. a-la Carrillo”) The function u verifies T 0 ∀k ∈ R ∀ξ ∈ D([0, T ) × Ω) + such that ξ| Σ\Σ ± (k) = 0 Ω −(u − k) ± ξt − q ± (u, k) · ∇ξ − (u0 − k) Ω ± ξ(0, ·) ≤ 0.
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Generalizing the Bardos-LeRoux-Nédélec boundary condition for ...

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