The shock formation and optimal regularities of the resulting shock curves for 1D scalar conservation laws
The shock formation and optimal regularities of the resulting shock curves for 1D scalar conservation laws
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DOI：
10.1088/1361-6544/ac4151
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发表时间：
2021-03
期刊：
影响因子：
1.7
通讯作者：
Yin Huicheng;Zhu Lu
中科院分区：
文献类型：
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作者：
Yin Huicheng;Zhu Lu
双曲型守恒律方程激波的形成和激波面的展开是非线性偏微分方程的基本问题。本文研究了一维守恒律方程f（u）= 0在光滑初值u（0，x）= u 0（x）下激波的形成和激波曲线的最佳拟合.如果u 0（x）∈C1（R）且f（u）∈C2（R），则已知当min g′（x）0（这被称为一般的非退化条件），然后由定理2 Lebaud（1994 J.Math.PuresAppl.9523 -565），弱熵解u与激波曲线x = φ（t）∈ C2 [T*，T* + φ）一起从爆破点开始（T*，x* = x 0 + g（x 0）T*）可以局部构造.当不满足一般的非退化条件时，即当x 0是g′（x）的一个局部极小点，使得g″（x 0）=g（3）（x 0）=.=g（2 k 0）（x 0）=0但0$？g（2k 0 +1）（x 0）>0，或g（k）（x 0）= 0，我们将研究激波曲线x = φ（t）的激波形成和最佳正则性，同时对u在爆破点（T*，x*）附近的行为给出一些精确的描述.我们的主要目的是证明：在爆破点附近，无论初始数据是有限阶退化还是无限阶退化，激波都真实地出现;激波解的最优阶数和由此得到的激波曲线与初始数据的退化程度有显式的关系。
The study on the shock formation and the regularities of the resulting shock surfaces for hyperbolic conservation laws is a basic problem in the nonlinear partial differential equations. In this paper, we are concerned with the shock formation and the optimal regularities of the resulting shock curves for the 1D conservation law ∂ t u + ∂ x f(u) = 0 with the smooth initial data u(0, x) = u 0(x). If u0(x)∈C1(R) and f(u)∈C2(R) , it is well-known that the solution u will blow up on the time T*=−1ming′(x) when min g′(x) 0 (which is called the generic nondegenerate condition), then by theorem 2 of Lebaud (1994 J. Math. Pures Appl. 9 523–565), a weak entropy solution u together with the shock curve x = φ(t) ∈ C 2[T*, T* + ɛ) starting from the blowup point (T*, x* = x 0 + g(x 0)T*) can be locally constructed. When the generic nondegenerate condition is violated, namely, when x 0 is a local minimum point of g′(x) such that g″(x0)=g(3)(x0)=…=g(2k0)(x0)=0 but 0$?> g(2k0+1)(x0)>0 for some k0∈N with k 0 ⩾ 2; or g (k)(x 0) = 0 for any k∈N and k ⩾ 2, we will study the shock formation and the optimal regularity of the shock curve x = φ(t), meanwhile, some precise descriptions on the behaviors of u near the blowup point (T*, x*) are given. Our main aims are to show that: around the blowup point, the shock really appears whether the initial data are degenerate with finite orders or with infinite orders; the optimal regularities of the shock solution and the resulting shock curve have the explicit relations with the degenerate degrees of the initial data.