On Kolmogorov Entropy Compactness Estimates for Scalar Conservation Laws Without Uniform Convexity
On Kolmogorov Entropy Compactness Estimates for Scalar Conservation Laws Without Uniform Convexity
复制标题
无一致凸性标量守恒定律的柯尔莫哥洛夫熵紧度估计
DOI：
10.1137/18m1198090
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
K. Nguyen
中科院分区：
文献类型：
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作者：
F. Ancona;O. Glass;K. Nguyen
在标量守恒律u t + f（u x）= 0，\qquad t\geq 0，x\in\mathbb{R}，$$具有一致严格凸通量f$的情形下，在DLG，AON 1中建立了关于在固定时间t>0$上估计的熵弱解集的定量紧性估计，其初始数据具有一致有界支撑并且在${\bf L}^\infty$的有界子集中变化。这些估计反映了这些非线性方程熵弱间断解的不可逆性特征。我们在这里提供了一个扩展这样的估计的情况下，标量守恒律的光滑通量函数f$，要么是严格（但不一定是一致的）凸或有一个单一的拐点与多项式退化。
In the case of scalar conservation laws $$ u_{t} + f(u)_{x}~=~0,\qquad t\geq 0, x\in\mathbb{R}, $$ with uniformly strictly convex flux $f$, quantitative compactness estimates - in terms of Kolmogorov entropy in ${\bf L}^{1}_{loc}$ - were established in~\cite{DLG,AON1} for sets of entropy weak solutions evaluated at a fixed time $t>0$, whose initial data have a uniformly bounded support and vary in a bounded subset of ${\bf L}^\infty$. These estimates reflect the irreversibility features of entropy weak discontinuous solutions of these nonlinear equations. We provide here an extension of such estimates to the case of scalar conservation laws with a smooth flux function $f$ that either is strictly (but not necessarily uniformly) convex or has a single inflection point with a polynomial degeneracy.