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22), we have − − H (x − p − y, t − τ ; −λ p ) + H (x − y, t − τ ; −λ p ) 1 = O(1)(t − τ )− 2 exp − 2 ((1 + O(1)ε)|x − y| + λ− p (t − τ )) 4(t − τ ) = O(1)H (x − y, t − τ ; −λ− p , µ)e −ε2 (t−τ )/C for some constant C > 0. 17). 20) is treated similarly. We introduce the notation + |λ p | ≡ min{λ− p , −λ p } > 0. 7. Let j be any nonnegative integer, and µ > 1 be a constant. 22) dj ∓ ± j+1 −|λ p ||x|/µ ρ ∓ (x; |λ∓ e . 23) Proof. 17). 23). 22), − + |ρb− (x; λ− p ) − ρb (x; |λ p |)| − λp x − + − e−λ p x = ρb− (x; λ− p )ρb (x; −λ p ) e + − λp x − + = O(1)ρb− (x; λ− + e−λ p x ε2 |x| = O(1)εe−|λ p ||x|/µ .
The idea can be extended to systems with physical viscosity in a straightforward manner. The details of construction, however, are much more involved. The results on these systems, including the Navier-Stokes equations and the equations for magnetohydrodynamics, of fundamental solutions and nonlinear stability of shock waves, are written in a forthcoming research monograph, [LZ3]. Results on stability of constant states and large time behavior for partially dissipative systems can be found in [Ze1,Ze2,LZ1,SZ,LZ2,Ze3,Ze4].
For this we introduce two sets of partition functions. The first pair of partition functions are for the transversal fields. 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| < .