I want to solve the following PDE for $c:\mathbb{R}^+\times\mathbb{R}^n\to\mathbb{R}$,
$${\frac {\partial c}{\partial t}}=\frac12\Delta c-{\vec {v}}\cdot \nabla c,$$
where $\vec{v}$ is a vector in $\mathbb{R}^n$ that might depend on $x$.
I looked on wikipedia and realised that it is called a convection-diffusion equation. I don't know how to solve it. I'd like to know if it is possible to solve it exactly (and how?), or is it only possible to solve it numerically?
Thank you very much!
Fourier transform in $\vec x$. Then, if you use $\hat f(\vec k) = \int f(\vec x) e^{-i \vec k \cdot \vec x} \, d^n x$, you get $$\partial_t \hat c(t, \vec k) = -\frac12 k^2 \hat c(t, \vec k) + \vec v \cdot i\vec k \hat c(t, \vec k) = \left(-\frac12 k^2 + \vec v \cdot i\vec k \right) \hat c(t, \vec k).$$
Thus, $$\hat c(t, \vec k) = \hat c(0, \vec k) \exp \left(t (-\frac12 k^2 + \vec v \cdot i\vec k) \right).$$
Then you "only" have to inverse-transform this.