# Stability of Multi-Dimensional Shock Fronts: A New Problem by Andrew Majda

By Andrew Majda

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Extra info for Stability of Multi-Dimensional Shock Fronts: A New Problem for Linear Hyperbolic Equations (Memoirs of the American Mathematical Society)

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7) involve a quantized version of the argument in [6J. 21), and integrate over x N to derive I O O Re dW A 0 (W, R(~) dX ~ N N-l OO + Re Re J: I 0 (W, (W, R((AN R L A. + ~~)-l ( j=l (w, (w, R(w)) I ~=O d~. 27) J (;~)w) ~I + (n + B)w)) d~ (w, Rw) A I ;;;. 2, Proposition N-l + ~ )-l( \' dW dW (n + B)w) j~l Aj dX + + j at 'I A-l, where p(XN' x' Ixl + It perties I t) ~RO. 17) to obtain that provided > C2. 7) provided --) vj,k' vj,k .

17) is violated £Q. 18) is valid. 22). 1 follows easily. d a 2 ~ 0, this can IIvII ~. 19) that + + a Lv IF+,n 120 + IF-,n I~ + (g}20 ,n is finite. 13) on the coefficients guarantees of unity systems which flatten the boundary. 11), by utilizing ~k' locally to a half-space a standard partition where supp ¢~ [-2, 2J. J , kilO , T) ~ C ( 2 IF+ j,k 1 O,T) 1 Provided that the constants, Cl + I 2 - Fj,k 1 O,T) T) and C , 2 depend only upon the quantities appearing in (l~. 1. 11) built from L+ a F. J, k Ixl > RO' F.

10) so that it is sufficient to consider 1/2 (QE) dp per- I + P Xl > 0 + + Also, (w , w , p) 2 1 by (the (w~,w{, p') . £L, dW~ __1 + (W+ - (J) __1 + + _1_+ w 1 2 dX dt dX. 2 1 i for 0 + dX. 21) . 27) >0 , except at the single 0 . 25), this factor. 1. £. '> d'- 0 I (c al - 2 + 2 + (wI - 0) ) + -2 c(w l points where s ~ 0, s ~ w(w~ - 0) , W2Jl < l pJ P p - 0) if and only if I+ P + (w+ 1 0)2 + W( w 1 - 0) + l w(w this directly but we omit the tedious is violated. 17) is violated £Q. 18) is valid. 22).