一维非线性严格双曲型守恒律组的SBV正则性
SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension
摘要：
我们证明了如果$t \mapsto u(t) \in \mathrm {BV}(\R)$是具有真正非线性特征场$N \times N$的严格双曲守恒律方程组的熵解，则直到可数的次数集$\{t_n\}_{n \in \mathbb N}$，函数$u(t)$在$\mathrm {SBV}$中，即它的分布导数$u_x$是没有Cantorian部分的测度。证明是基于分解的$u_x(t)$波属于的特征家庭\[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N, \]和平衡的连续/跳跃部分的措施$v_i$在区域内的特点。为此，引入了一种新的相互作用测度$μ_{i,\jump}$，控制测度$v_i(t)$中原子的产生。证明的主要论点是，对于所有$t$的Cantorian部分的$v_i$不为0，要么Glimm功能有一个向下的跳跃，或有一个抵消波或措施$μ_{i,\mathrm{jump}}$是积极的。
英文摘要：
We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \[ u_t + f(u)_x = 0, \] then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N, \] and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure $μ_{i,\jump}$ is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $μ_{i,\mathrm{jump}}$ is positive.
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