UMD PDE Qualifying Exams/Jan2011PDE

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Problem 1[edit]

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.