具有可积无界源的平衡律
Balance laws with integrable unbounded sources
摘要：
本文考虑$n\times n$严格双曲平衡律组$$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | λ_i(u)| \geq c > 0 {for all} i\in \{1,...,n\}, \|g(x,\cdot)\|_{\mathbf{C}^2}\leq \tilde M(x) \in L1, {array}. $$的Cauchy问题，其中每个特征场是真非线性的或线性退化的。假设$\|g(x,\cdot)\|_{\mathbf{C}^1}$和$\|u_o\|_{BV(\reali)}$的$\mathbf{L}^1$范数足够小，我们证明了有界全变差的整体熵解的存在唯一性，将[1]的结果推广到无界（在$L^\infty$中）源.此外，我们将这一结果应用于具有不连续横截面的管道中的流体流动，证明了基础半群的存在性和唯一性。
英文摘要：
We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | λ_i(u)| \geq c > 0 {for all} i\in \{1,...,n\}, \|g(x,\cdot)\|_{\mathbf{C}^2}\leq \tilde M(x) \in L1, {array}. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the $\mathbf{L}^1$ norm of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{BV(\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in $L^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
