UMD PDE Qualifying Exams/Jan2011PDE

Problem 1

Consider the conservation law $u_{t}+f(u)_{x}=0,\quad {\text{ in }}\mathbb {R} \times (0,\infty ),$ where $f\in C^{1}(\mathbb {R} )$. (a) Define an integral solution of the PDE (b) Derive the jump (Rankine-Hugoniot) condition satisfied by a piecewise smooth integral solution $u$ across a $C^{1}$ curve where this $u$ has a discontinuity. (c) Find an integral solution to the PDE when $f(u)=u^{2}+u$ with $u(x,0)=1$ if $x<0,$ $u(x,0)=-3$ if $x>0$.