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What's a measure valued solution of a PDE?

For instance the Fokker-Planck equation \begin{align} \partial_t\mu_t+\sum_i\partial_i(b_i\mu_t)-\frac{1}{2}\sum_{ij}\partial_{ij}(a_{ij}\mu_t)=0 \end{align} it says that for a measure $\mu=\mu_(t,x)=\mu_t(x)$ being a solution of the above equation means \begin{align} \frac{d}{dt}\int_{\mathbb{R}^N}\phi(x)d\mu_t(x)=\int_{\mathbb{R}^N}\left(\sum_ib_i(t,x)\partial_i\phi(x)+\frac{1}{2}\sum_{ij}a_{ij}(t,x)\partial_{ij}\phi(x)\right)d\mu_t(x) \end{align} How do we get this? Also if compare this with the weak formulation, What's the connection between a measure valued solution and a distributional solution?

Many thanks!

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What is $\phi(x)$? Is it a test-function or something else? –  Pantelis Sopasakis Aug 28 '12 at 10:20

1 Answer 1

up vote 2 down vote accepted

Here's some motivation for the definition: If we have a (positive) classical solution f(t,x) define the measure $\mu(t,x)$ on $\mathbb{R}^N$ by $\mu(t,x)(A) = \int_{A} f(t,x)dx$, so that $d\mu_t = f(t,x)dx$. Multiplying by a test function $\phi(x),$ differentiating in $t$ and integrating by parts gives us exactly the above formula.

Measure-valued solutions are convenient to use in many situations, as far as I can tell because they simplify the existence theory for PDE. For another example, one can define measure-valued solutions to the Monge-Ampere equation $\det D^2u = \nu$ by requiring that $u$ is a convex function with $|Du(A)| = \nu(A)$, where $Du(A)$ is the collection of slopes of supporting hyperplanes to $u$ in the set $A$. In this setting, the motivation for the definition is the area formula, $|Du(A)| = \int_{A} \det D^2u.$

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