The Burgers equation is the following:
Here can be regarded as the (scalar) velocity of a fluid and can be regarded as the viscosity. This equation can be regarded as the simplified version of the Euler’s equation without the pressure term.
This equation is non-linear. Instead it is semilinear (the highest order term has coefficient that depends on only. In fact, it is constant). It turns out that this equation is related with the 1-D (linear!) heat equation
via the so-called Hopf-Cole transformation. It is well-known that the heat equation can be solved by convolving the initial function with the heat kernel. Therefore we can actually solve the Burgers equation explicitly, although the result is a bit complicated. I will record the result here.
First of all, we integrate (1) w.r.t. the first variable of and define . From this we can get
This is still non-linear. It turns out that we can remove the non-linearity by using logarithm. With some hindsight, let
where is to be chosen. Then (2) becomes
So to remove the non-linearity, set , i.e.
Then (2) becomes
It is well-known that the heat equation
has unique solution
From this, and after some non-trivial work, we have
Theorem 1 (Hopf) Under some technical condition, the function
solves (1) with the initial condition , where