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By M. Aizenman (Chief Editor)

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Then −1 + ε2/3 | ln ε|1/3 δ, √ 3 (i) If δ¯ ≤ 21 9 κ 2 , then ε−4/3 | ln ε|−2/3 min E → 21 κ −2 δ¯2 , √ √ √ 3 3 (ii) If δ¯ > 1 3 9 κ 2 , then ε−4/3 | ln ε|−2/3 min E → 9 δ¯ − 9 κ 2 , 2 2 4 as ε → 0. 3 says that the energy of the minimizers of the diffuse interface energy E behaves asymptotically the same as that of the sharp interface energy E in the limit ε → 0. In particular, the transition to non-trivial minimizers occurs asymptotically at the same values of u¯ for ε 1. 3 are based on a number of propositions established in Secs.

Equ. : Stability of undercompressive viscous shock waves. J. Differ. Equ. : Stability of isentropic Navier-Stokes shocks in the high-Mach number limit. Commun. Math. Phys. : Spectral stability of ideal-gas shock layers. Arch. Ration. Mech. Anal. : Multidimensional spectral stability of large-amplitude Navier-Stokes shocks. : Multi-dimensional diffusion waves for the navier-stokes equations of compressible flow. Indiana Univ. Math. J. : Pointwise decay estimates for multidimensional navier-stokes diffusion waves.

Stability of undercompressive viscous shock waves. J. Differ. Equ. : Stability of isentropic Navier-Stokes shocks in the high-Mach number limit. Commun. Math. Phys. : Spectral stability of ideal-gas shock layers. Arch. Ration. Mech. Anal. : Multidimensional spectral stability of large-amplitude Navier-Stokes shocks. : Multi-dimensional diffusion waves for the navier-stokes equations of compressible flow. Indiana Univ. Math. J. : Pointwise decay estimates for multidimensional navier-stokes diffusion waves.

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