Reduction on characteristics for continuous solutions of a scalar balance law
Reduction on characteristics for continuous solutions of a scalar balance law
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标量平衡律连续解的特征约简
DOI：
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发表时间：
2014
期刊：
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通讯作者：
L. Caravenna
中科院分区：
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作者：
G. Alberti;S. Bianchini;L. Caravenna
考虑平衡方程<$tu（t，x）+<$x [f（u（t，x））] = g（t，x）的连续解u，其中f是C类的，源项g是有界的.当f是一致凸的时，连续性改进为Hölder连续性，但一般来说它不是更正则的。我们主要在联合工作[5，1]的基础上讨论了特征上的常微分方程的约化问题。我们在这里提供了局部Lipschitz正则性结果，它在f ′（u）f ′′（u）6= 0的区域内成立，并且只在自治源g = g（x）的较简单情况下成立，但对于解u（t，x），它可能依赖于时间.这对应于常微分方程组{ γ今（t）= f ′（u（t，γ（t），ddt u（t，γ（t））= g（γ（t））的局部Lipschitz正则性结果。
We consider continuous solutions u to the balance equation ∂tu(t, x) + ∂x [f(u(t, x))] = g(t, x) , where f is of class C and the source term g is bounded. Continuity improves to Hölder continuity 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 Lipschitz regularity results holding in the region where f ′(u)f ′′(u) 6= 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 Lipschitz regularity result, in that region, for the system of ODEs { γ̇(t) = f ′(u(t, γ(t))) , d dt u(t, γ(t)) = g(γ(t)) .