Equation of the type evolution is one important topic in mathematics. Today, I read some very theoretical basis of this subject.
∂tf=Δf(x)+a(x)⋅∇f(x)+c(x)f(x)+∫Rdb(y,x)f(y)dy
with the condition that a∈W1,∞,c∈L∞,b∈L2(Rd×Rd).
Although there are many equations (maybe of types degenerate), but this model describes the equation of diffusion, of branching, of transport and also of mean-field, so it is already very large. Of course, we will talk about the existence, uniqueness of this equation.
The main frame to talk about the evolution equation is to design a Hilbert space H and a Banach subspace V⊂H, the advantage is to use the duality of Hilbert space to get
V⊂H=H′⊂V′.
so we can talk about the weak solution of the equation and we use |⋅|,‖ to represent respectively the norm in H and V.
One propose a good condition, similar to that of Lax-Milgram. If we denote by L the generator, then we requires that
(1)L : V \rightarrow V' is bounded.
(2)L is coercive + dissipative that \langle Lg, g \rangle \leq - \alpha \|g\|^2 + b|g|^2 for every g \in V.
This condition gives an inequality
|g(T)|^2_H + 2 \alpha \int_0^T \|g(s)\|_V^2 ds \leq e^{2bT}|g_0|^2
at first correct for a g more regular and then we can pass to general function by approximation. This is the energy inequality, it says a lot of things and the most important the uniqueness and make sure that the solution stays always in the function space g \in L^{\infty}(0, T; H) \cap L^2(0,T;V) \cap H^1(0, T; V').
(One remark : the dissipative part just says the Cauchy-Liptchitz condition. )
Later, we use a theorem : L^2(0,T;V) \cap H^1(0, T; V') \subset C([0,T];H). This makes sense when we do formally g' and then we can manipulate as if it is regular.
The existence theory is a variant theorem of Lax-Milgram theory. When we discretize the problem, it becomes a variant elliptic equation in small discrete steps. Then, we do a sub-sequence of weak limit to get the solution of the equation. .
