Let v E C1(R) and consider the equation

dt(u) + dx(uv) = 0, x E R, t>0,

u(x,0) = u0(x), x E R,

We assume that v satisfies v(x) > 0, v’ (x) > 0 for all x E R.

Prove that sup|u(x,t)| < sup|u0(x)| for all t > 0

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- February 9th 2010, 04:20 AMRazsterMcCoilAdvection Equation
Let v E C1(R) and consider the equation

dt(u) + dx(uv) = 0, x E R, t>0,

u(x,0) = u0(x), x E R,

We assume that v satisfies v(x) > 0, v’ (x) > 0 for all x E R.

Prove that sup|u(x,t)| < sup|u0(x)| for all t > 0