Appendix to Paper : a Geometric Level - Set Formulation of a Plasma - Sheath Interface

In this paper, we present appendices employed in the paper ”A geometric level-set formulation of a plasma-sheath interface” by the authors. Appendix A. Local existence of sheath solutions In this appendix, we present a series of a priori estimates for the approximate solutions constructed in Step 0 Step 7 in Section 7 and then give the proof Theorem 7.3. A.1. Basic a priori estimates. In this part, we give a priori estimates for the approximate solutions constructed in Step 0 Step 7. A.1.1. A priori estimates for Step1 . In Lemmas A1-A3, we will give a proof of the existence, uniqueness and regularity for n as given in Step 1 of Section 7.2. We first consider the equation for a characteristic curve. For given (x, t), (A.1) ∂sχ(s, t,x) = v(χ(s, t,x), s), χ(t, t,x) = x, 0 ≤ s ≤ T. In what follows, we will use calculus type estimates for the Hölder seminorm. For fi ∈ C(Λ̄s(T ;v)) i = 1, 2, we have [[f1f2]]0,γ ≤ [[f1]]0,γ |||f2|||0 + |||f1|||0[[f2]]0,γ , (A.2) [[e ]]0,γ ≤ e 0[[f ]]0,γ . (A.3) Here [[·]]0,γ and ||| · |||0 denote the Hölder and esssup norms defined on the same space-time region. In the following Lemma, we use simplified notation for balls in R2: B1 := B(0, rb + 3K0δ T0) and B2 := B(0, rb + 6K0δ T0). Lemma A.1. There exists a sufficiently small constant T0 > 0 and a unique solution χ to the equation (A.1) satisfying the following estimates: For 0 < T ≤ T0, v ∈ B(T ), (1) The forward characteristic curve χ(s, 0,x), s ≥ 0, x ∈ Ωs(0;v) ⊂ (B1 − Ω0) hits the target boundary ∂Ω0 and the ion-density in the region Λ 1 s(T ;v) is given by n(χ(t, 0,x), t) = n0(x) exp ( − ∫ t 0 (∇ · v)(χ(s, 0,x), s)ds ) , x ∈ Ωs(0;v). (2) χ(s, t,x) ∈ C1,γ([0, T ] × [0, T ]× R2) and sup s,t,x max i,j=1,2 |∂xjχ| ≤ 2. 1 2 MIKHAIL FELDMAN, SEUNG-YEAL HA, AND MARSHALL SLEMROD (3) Suppose that vi → v in C1,γ(Λ̄(T )) and let χi and χ be the characteristic curves corresponding to vi and v respectively. Then for (x, t) ∈ Λs(T ;v), χi(·, t,x) → χ(·, t,x) in C([0, T ]). (4) α(x, t) := χ(0, t,x) is Lipschitz continuous in (x, t) ∈ Λ(T ) with a Lipschitz constant 4, i.e. |α(x, t) − α(y, s)| ≤ 4|(x, t) − (y, s)|. Proof. (i) It follows from the dissipative condition (D2) in the definition of B(T ), we have v(x, t) · x ≤ −0 2 |x|, (x, t) ∈ (B2 − Ω0)× [0, T ]. Then we have for (x, t) ∈ (B2 − Ω0)× [0, T ], d ds |χ(s, t,x)| = 2〈v(χ(s, t,x), s),χ(s, t,x)〉 ≤ −η0|χ(s, t,x)|. Here 〈·, ·〉 denotes the standard inner product in R2. Hence the characteristic χ(s, t,x) satisfies |χ(s, 0,x)| ≤ e η0s 2 |χ(0, 0,x)| = e η0s 2 |x|. So χ(s, 0,x) has decreasing magnitude and must hit the target at some positive s. Let T ≤ T0 and we define the subregions Λs(T ;v),Ωs(T ;v) of Λ(T ) and Ωs(0) as in Step 1 of Section 7.2.1. Then the characteristic curve χ(s, 0,x), (x, 0) ∈ Ωs(0) × {t = 0} hits the target boundary ∂Ω0 and will provide the ion density n at the target boundary, i.e., n(χ(t, 0,x), t) = n0(x) exp ( − ∫ t 0 (∇ · v)(χ(s, 0,x), s)ds )