Regularity of PDEs, general question (Example: Kolmogorov equation)

Let us consider a differential operator $L$ and the PDE $$Lu=0$$ with some boundary conditions (assume Neumann conditions). For example, Kolmogorov's equation $$(1) \quad Lu(t,x) = \partial_t u(t,x) + b(t,x) \partial_x u(t,x)+ \frac{1}{2}\partial_x^2 u(t,x) =0, \quad u(T,x) = g(x), \quad t\in [0,T].$$

Then one can consider the adjoint equation, namely $$L^\ast u =0$$ which in the example above is the Fokker-Planck equation $$(2) \quad L^\ast u(t,x) = -\partial_t u(t,x) -\partial_x [b(t,x)u(t,x)]+\frac{1}{2} \partial_x^2 u(t,x), \quad u(0,x)=g(x), \quad t\in [0,T].$$

My question is: Whatever regularity properties you obtain for $u$ from (1), are these properties transferable to (2)?. In other words, studying (1) is equivalent to studying (2) when it comes to well-posedness and regularity of the solution? I am specially interested in this matter for this particular PDE (Kolmogorov)

Thanks for any feedback or ideas!

Solution 1:

You first need to define a suitable Hilbert space in which the states evolve. The state space should be chosen in such a way that for a particular boundary condition the PDE yields one and only one solution!

The PDE here consists of a linear operator and a nonlinear one. Apply Lumer-Phillips theorem to make sure the linear part is the generator of a (contraction) semi-group. Then look at the nonlinear part, there are many scenarios:

• A Frechet differentiable nonlinearity

• A locally Lipschitz nonlinearity

• A locally Lipschitz nonlinearity on the fractional powers of the state space (only if the semigroup is analytical, so-called parabolic PDEs)

Once your are done with the well-posedness and uniqueness. You can define the adjoint system to be the system along which the inner-product of adjoint state with original state remains constant.

For a linear system everything comes straightforward, and many properties carry over to adjoint system. However, for a nonlinear PDE, the existence and uniqueness result cannot be transferred.