论文标题
关于平滑解决方案的爆炸机制，具有大量初始数据的严格双曲线系统
On the blowup mechanism of smooth solutions to 1D quasilinear strictly hyperbolic systems with large initial data
论文作者
论文摘要
对于第一阶1d $ n \ times n $ quasilIrinear严格双曲系统$ \ partial_tu+f（u）\ partial_xu = 0 $ with $ u（x，0）= \ varepsilon u_0（x）$ C_0^2（\ Mathbb r）$，当至少一个$ f（u）$的特征值是真正的非线性时，众所周知，在有限的爆炸时间$ t _ {\ varepsilon} $，衍生物$ \ partial_ {对于1D标量方程或$ 2 \ times 2 $严格双曲系统（对应于$ n = 1，2 $），如果平滑的解决方案$ u $在有限的时间内爆炸，则可以很好地理解爆炸机制（即，仅发生$ \ partial_ {t，x，x} u $的爆炸）。在本文中，对于$ n \ times n $（$ n \ geq 3 $），严格的双曲线系统带有一类大型初始数据，我们关注有限的爆炸时间的平滑解决方案$ u $的爆炸机制，以及$ \ partial_ {t，x} u $ ablup点附近的$ \ partial_ {我们的结果基于$ u $沿不同特征方向的有效分解，适当的调制坐标和全球加权能量估计。
For the first order 1D $n\times n$ quasilinear strictly hyperbolic system $\partial_tu+F(u)\partial_xu=0$ with $u(x, 0)=\varepsilon u_0(x)$, where $\varepsilon>0$ is small, $u_0(x)\not\equiv 0$ and $u_0(x)\in C_0^2(\mathbb R)$, when at least one eigenvalue of $F(u)$ is genuinely nonlinear, it is well-known that on the finite blowup time $T_{\varepsilon}$, the derivatives $\partial_{t,x}u$ blow up while the solution $u$ keeps to be small. For the 1D scalar equation or $2\times 2$ strictly hyperbolic system (corresponding to $n=1, 2$), if the smooth solution $u$ blows up in finite time, then the blowup mechanism can be well understood (i.e., only the blowup of $\partial_{t,x}u$ happens). In the present paper, for the $n\times n$ ($n\geq 3$) strictly hyperbolic system with a class of large initial data, we are concerned with the blowup mechanism of smooth solution $u$ on the finite blowup time and the detailed singularity behaviours of $\partial_{t,x}u$ near the blowup point. Our results are based on the efficient decomposition of $u$ along the different characteristic directions, the suitable introduction of the modulated coordinates and the global weighted energy estimates.
