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By M. Aizenman (Chief Editor)

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7), hence the width of the shock layer is 1/ε. We define partition functions ρa∓ as mollified Heaviside functions with the same width. 13) where J (x; ε) = ε ⎧ ⎪ ⎨ K exp − ⎪ ⎩0 1 1 − (εx)2 1 ε 1 if |x| ≥ ε if |x| < is the mollifier. 13) we have ⎧ ⎪ ⎪ ⎪0 ⎪ ⎪ ⎨ + ρa (x) = K ⎪ ⎪ ⎪ ⎪ ⎪ ⎩1 εx −1 J (x; ε) d x = 1. exp − 1 1 − y2 dy 1 if x ≤ − ε 1 if |x| < . 14) We define ⎧ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨ ρa− (x) ≡ 1 − ρa+ (x) = K ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 1 ε 1 if |x| < . 1. 15) O(1)ε, 1 ε = O(1)ε j e−|λ||x| . 1 ε Proof. 15). -P. Liu, Y.

3 we derive estimates on the fundamental solution. In particular, we assess the truncation error. In Sect. 4 we give results on wave interaction. The estimates in Sects. 3 and 4 are then applied to the a priori analysis in Sect. 2. 23) we let λi∓ ≡ λi (u ∓ ), li∓ ≡ li (u ∓ ), ri∓ ≡ ri (u ∓ ), etc. Throughout this paper we use C > 0 to denote a sufficiently large constant, and any O(1) is a function bounded uniformly with respect to δ0 , ε, x, t, y, τ and any other independent variables. 2. 21).

2 says the φ(x) − u ∓ and its derivatives are along r p (φ) in the leading term. This is a reasonable assumption, true for physical systems such as Navier-Stokes equations. 5a) is true for j = 0. 3. Let j ≥ 0 be an integer. Let µ > 1 be a fixed constant. 7b) − λ p x/µ dj if x ≤ 0 j+1 e = O(1)ε , φ(x) + x/µ λ j p dx e if x ≥ 0 j ≥ 1. 3 applies to systems with physical viscosity, and is proved in [LZ3]. Next we have a few lemmas related to cancellations. 4. Let j be any nonnegative integer and K > 0 be a constant.

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Communications In Mathematical Physics - Volume 290 by M. Aizenman (Chief Editor)

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