Solving KPZ/Navier-Stokes Type Equations Using Distribution Dependent SDEs

By using a new type distribution dependent stochastic differential equation (SDE), the existence, uniqueness and non-explosion are derived for nonlinear PDEs of type ∂tut = Ltut +  Ft(·, ut , ∇ut) · ∇ ut + gt for u : [0, ∞) × R d → R m, where for each t ≥ 0, Lt is a singular second order differential operator, Ft : R d ×R m ×R d⊗m → R d is bounded in x ∈ R d and local Lipschitz continuous in the other two components, and gt : R d → R m with kgtk∞ locally integrable in t. In particular, KPZ and Navier-Stokes type equations are included.