Pointwise convergence to shock waves for viscous conservation laws

✍ Scribed by Tai-Ping Liu

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 459 KB
• Volume: 50
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199711)50:11<1113::aid-cpa3>3.0.co;2-d

✦ Synopsis

We are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws. It is shown that, besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and Burgers equations, a perturbation also gives rise to algebraically decaying terms, which measure the coupling of waves of different characteristic families. Our technique is a combination of time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions. The pointwise estimates yield optimal L p convergence of the perturbation to the shock and diffusion waves, 1 ≤ p ≤ ∞.

The new approach of obtaining pointwise estimates based on the Green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks. These are being explored elsewhere.