Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 4 / a

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- Solution a

Solution

Take the real

L^{2}

inner product of the equation with

u

. The homogeneous Dirichlet boundary conditions and integration by parts give

\frac{1}{2} \frac{d}{d t} ∥ u (t) ∥_{L^{2}}^{2} = \int_{- 1}^{1} u u_{xx} d x + α \int_{- 1}^{1} u u_{x} d x = - \int_{- 1}^{1} ∣ u_{x} ∣^{2} d x + \frac{α}{2} [u^{2}]_{- 1}^{1} = - ∥ u_{x} ∥_{L^{2}}^{2} \leq 0.

Thus

∥ u (t) ∥_{L^{2}} \leq ∥ u_{0} ∥_{L^{2}}

, and applying the same estimate to the difference of two solutions gives continuous dependence on the initial data. The constant-advection term is skew-symmetric under these boundary conditions and contributes no energy. Hence

the problem is well posed in L^{2} for every α \in R .

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