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Additional info for Communications In Mathematical Physics - Volume 290
Direct calculation yields 1 + eε0 y H (x − y, t − τ ; ε0 ) 1 + eε0 x 1 + e−ε0 y = H (x − y, t − τ ; −ε0 ) 1 + e−ε0 x = ρbσ (x; ε0 )H (x − y, t − τ ; −σ ε0 ). 35) Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile 37 µ > 0 and λ being constants. 21). 34). That is, the approximate fundamental solution recovers the exact one in the case of the Burgers equation. 29) is the right one. 14. For that we will need the following lemma. 6. 36) where |λ| > 0 is a parameter. Proof. 36) is obtained by direct calculation.
21). That is, our first consideration of the shift is to guarantee the continuity of characteristic curves when crossing the shock. In the above intuitive discussion we have basically regarded the shock as with zero width. For the viscous shock wave with strength ε, the width is O(1/ε), and it is crucial to look into the shock layer. 28) no longer collapses into a single term, even when x > 0 is large. It is an interpolation of two heat kernels, one starting at (x, t) with speed λi+ , and the other starting at (xi− , t) with speed λi− .
Let ε be sufficiently small. 26) d j1 ∓ ∂ j2 ± ρb (x; |λ± |) j H x ± p p − y, t − τ ; −λ p j dx 1 ∂y 2 = O(1)e−|λ p ||x|/µ ε j1 (t − τ )− j2 +1 2 −ε H x − y, t − τ ; λ∓ p,µ e 2 (t−τ )/C , where C > 0 is some constant. Proof. 10). 22) as well. 10. Let j1 and j2 be any nonnegative integers, and µ > 1 be a constant. 28) j2 d j1 ∓ ± ∂ ± ρ (x; |λ |) H x± p p − y, t − τ ; −λ p d x j1 b ∂ y j2 j2 −[ε = O(1)ρb± (x; |λ p |)ε j1 (t − τ )− 2 H x − y, t − τ ; λ± p,µ e 2 (t−τ )+ε|x|]/C , where C > 0 is some constant.