Differentiability with respect to initial data for a scalar conservation law

Francois Bouchut

Description

Niveau: Supérieur, Doctorat, Bac+8
Differentiability with respect to initial data for a scalar conservation law Franc¸ois BOUCHUT Franc¸ois JAMES? Abstract We linearize a scalar conservation law around an entropy initial datum. The resulting equation is a linear conservation law with discontinuous coefficient, solved in the context of duality solutions, for which existence and uniqueness hold. We interpret these solutions as weak derivatives with respect to the initial data for the nonlinear equation. Proceedings of the seventh Conference on Hyperbolic Problems (Zurich 1998), M. Fey & R. Jeltsch, Eds., International Series of Numerical Mathematics, 129 Birkhauser, Basel, 1999, 113-118. 1 Introduction Consider the one-dimensional scalar conservation law ∂tu+ ∂xf(u) = 0, 0 < t < T, x ? R, (1) where f is a C1 convex function, provided with entropy admissible initial data u? ? L∞(R). Kruzˇkov's results [4] assert that the entropy solution u to (1) lies in L∞(]0, T [?R)?C(0, T ;L1loc(R)), and that the following contraction property holds: if u (resp. v) corresponds to the initial data u? (resp. v?), then for all R > 0 and any t > 0 ∫ |x|≤R |u(t, x)? v(t, x)| dx ≤ ∫ |x|≤R+Mt |u?(x)? v?(x)| dx, (2) where M

• cauchy problem


• weak differentiability


• usual weak


• entropy admissible


• unique µ ?


• dimensional perturbation


• oleinik's entropy


• indeed

Sujets

• Bouchut
• Cauchy problem
• Indeed.com

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 7
• Langue: English

Extrait