Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws
Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws
复制标题
多维标量守恒定律中非线性的正则化效应
DOI：
10.1007/978-3-540-76781-7_3
复制
发表时间：
2008
期刊：
影响因子：
12.8
通讯作者：
Michael Westdickenberg
中科院分区：
文献类型：
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作者：
Gianluca Crippa;F. Otto;Michael Westdickenberg
在这些注记中，我们将讨论标量多维守恒律，它是形式为<$u <$t+<$f· f（u）= 0 <$（t，x）∈ R+× Rn的一阶偏微分方程。（1）这里f：R−→ Rn是一个光滑映射，称为通量函数。解决方案u：R+× Rn−→ R可以被看作是一个守恒量：事实上，如果我们在一个光滑边界的有界集合Ω <$Rn上积分方程，并应用发散定理，我们可以看到Ω中u的总量随时间变化，这取决于f（u）穿过边界<$Ω的通量。这种解释激发了人们对这类方程的兴趣：自然界中的许多情况都是以
In these notes we will deal with scalar multidimensional conservation laws, which are first order partial differential equations of the form∂ u∂ t+∇· f (u)= 0∀(t, x)∈ R+× Rn.(1)Here f: R−→ Rn is a smooth map, which will be called the flux function. The solution u: R+× Rn−→ R can be viewed as a conserved quantity: indeed, if we integrate the equation over a bounded set Ω⊂ Rn with smooth boundary and we apply the divergence theorem, we see that the total amount of u in Ω changes in time according to the flux of f (u) across the boundary∂ Ω. This interpretation motivates the interest of this class of equations: many situations in nature are modeled on the