Abstract
We consider the Cauchy problem for a strictly hyperbolic, n × n system in one-space dimension: ut + A(u)ux = 0, assuming that the initial data have small total variation. We show that the solutions of the viscous approximations ut + A(u)ux = εu xx are defined globally in time and satisfy uniform BV estimates, independent of ε. Moreover, they depend continuously on the initial data in the L1 distance, with a Lipschitz constant independent of t, ε. Letting ε → 0, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where A = Df is the Jacobian of some flux function f : ℝn → ℝn, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws ut + f(u)x = 0.
Original language | English (US) |
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Pages (from-to) | 223-342 |
Number of pages | 120 |
Journal | Annals of Mathematics |
Volume | 161 |
Issue number | 1 |
DOIs | |
State | Published - Jan 2005 |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty