What's a measure valued solution of a PDE?

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?

• What is $\phi(x)$? Is it a test-function or something else? Commented Aug 28, 2012 at 10:20

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, $$|Du(A)| = \int_{A} \det D^2u.$$