Hyperbolic conservation laws with nonlinear diffusion and nonlinear dispersion
- Fujino, Naoki
- Yamazaki, Mitsuru
Publication date
September 2006
DOI Publisher
Elsevier Inc. ISSN
0022-0396 Citation count (estimate)
2Abstract
AbstractWe study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms ut+f(u)x=ε∂xb(∂xu)−δ∂x2((∂xu)2ℓ−1) (any ℓ⩾1), the flux function f(u) being mth order growth at infinity. It is shown that if ε, δ=δ(ε) tend to 0, then the sequence {uε} of the smooth solutions converges to the unique entropy solution u∈L∞(0,T∗;Lq(R)) to the conservation law ut+f(u)x=0 in Lk(0,T*;Lp(R))(k<∞,p<q). The proof relies on the methods of compensated compactness, Young measures and entropy measure-valued solutions. Some new a priori estimates are carried out. In particular, our result includes the convergence result by Schonbek when b(λ)=λ, ℓ=1 and LeFloch and Natalini when ℓ=1
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