UMD PDE Qualifying Exams/Jan2011PDE

Problem 1

Consider the conservation law $u_t+f(u{)}_x=0,\text{ in }ℝ\times (0,\mathrm{\infty }),$ where $f\in C^1(ℝ)$. (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 $x>0$.