Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 5 / 4 / a / ii

Table of contents
- Solution ii

Solution

Multiply the viscous scalar conservation law by

- u_{xx}

, rather than estimating

F^{''}

after differentiating. Integration by parts yields

\frac{1}{2} \frac{d}{d t} ∥ u_{x} ∥_{2}^{2} + ε ∥ u_{xx} ∥_{2}^{2} = \int F^{'} (u) u_{x} u_{xx} \leq M ∥ u_{x} ∥_{2} ∥ u_{xx} ∥_{2} \leq \frac{ε}{2} ∥ u_{xx} ∥_{2}^{2} + \frac{M ^{2}}{2 ε} ∥ u_{x} ∥_{2}^{2} .

Thus

d ∥ u_{x} ∥_{2}^{2} / d t \leq (M^{2} / ε) ∥ u_{x} ∥_{2}^{2}

. Applying the Gronwall inequality gives

∥ u_{x} (t) ∥_{2}^{2} \leq e^{M^{2} t / ε} ∥ u_{x} (0) ∥_{2}^{2}, C_{1} = M^{2} / ε .

Only the assumed bound on

F^{'}

is used; a global bound on

F^{''}

is not needed for this energy estimate.

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