We consider continuous solutions u to the balance equation ∂t u(t, x) + ∂x [f (u(t, x))] = g(t, x) f ∈ C 2 (R), g ∈ L∞ (R) for a bounded source term g. Continuity improves to H ̈lder continuity o when f is uniformly convex, but it is not more regular in general. We discuss the reduction to ODEs on characteristics, mainly based on the joint works [5, 1]. We provide here local regularity results holding in the region where f (u)f (u) = 0 and only in the simpler case of autonomous sources g = g(x), but for solutions u(t, x) which may depend on time. This corresponds to a local regularity result, in that region, for the system of ODEs γ(t) = f (u(t, γ(t))) ̇ d u(t, γ(t)) = g(t, γ(t)). dt
Reduction on characteristics for continuous solutions of a scalar balance law / Alberti, G; Bianchini, Stefano; Caravenna, L.. - 8:(2014), pp. 399-406. (Intervento presentato al convegno 14th International Conference devoted to Theory, Numerics and Applications of Hyperbolic Problems (HYP) tenutosi a Padova, Italy nel 2012).
Reduction on characteristics for continuous solutions of a scalar balance law
Bianchini, Stefano;
2014-01-01
Abstract
We consider continuous solutions u to the balance equation ∂t u(t, x) + ∂x [f (u(t, x))] = g(t, x) f ∈ C 2 (R), g ∈ L∞ (R) for a bounded source term g. Continuity improves to H ̈lder continuity o when f is uniformly convex, but it is not more regular in general. We discuss the reduction to ODEs on characteristics, mainly based on the joint works [5, 1]. We provide here local regularity results holding in the region where f (u)f (u) = 0 and only in the simpler case of autonomous sources g = g(x), but for solutions u(t, x) which may depend on time. This corresponds to a local regularity result, in that region, for the system of ODEs γ(t) = f (u(t, γ(t))) ̇ d u(t, γ(t)) = g(t, γ(t)). dtFile | Dimensione | Formato | |
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