### Scalar Non-linear Conservation Laws with Integrable Boundary Data

Fabio Ancona and Andrea Marson

Abstract: We consider the initial-boundary value problem for a scalar non-linear conservation law $$u_t [f(u)]_x=0,\qquad u(0,x)=\bar u(x),\qquad u(t,0)=\tilde u(t),\eqno(*)$$ on the domain $\Omega=\{(t,x)\in{\bf R}:t\geq 0,x\geq 0\}$.Here $u=u(t,x)$ is the state variable, $\bar u$, $\tilde u$ are integrable (possibly unbounded) initial and boundary data, $f$ is assumed to be strictly convex. We first derive an explicit formula for a solution of (*) which generalizes a result of Le Floch in the case where $\bar u\in{\bf L}^1,\,f(\tilde u)\in{\bf L}^1_{loc}$. Moreover we extend a contractive property and establish a comparison principle for integrals of solutions of (*). Next we study the problem (*) taking $\bar u\equiv 0$ and letting $\tilde u$ vary into a given set ${\Cal U}\subseteq{\bf L}^1_{loc}$ of boundary data regarded as admissible controls. In particular, we give a characterization of the set of attainable profiles at a fixed time $T>0$ and at a fixed point $\bar x>0$ when $${\Cal U}=\{\tilde u\in {\bf L}^1_{loc}({\bf R}^ ):f(\tilde u)\in{\bf L}^1_{loc}({\bf R}^ ), f^\prime(\tilde u)\geq 0\}.$$

Paper:
Available as PostScript
Title:
Scalar Non-linear Conservation Laws with Integrable Boundary Data
Author(s):
Fabio Ancona, <ancona@ciram3.ing.unibo.it>
Andrea Marson, <marson@bsing.ing.unibs.it>
Publishing information:
To appear on Nonlinear Analisys, Theory, Methods and Applications