Sums of generators and the Lie--Trotter formular (Q2743187)

The author considers the Cauchy problem NEWLINE\[NEWLINE{\dot u}(t)=(A+B)u(t),\quad t\geq 0; \quad u(0)=f,\tag \(*\) NEWLINE\]NEWLINE where \(A\) is a generator of a (well understood) strongly continuous semigroup \(T(t)\), and \(B\) is a perturbation operator in a Banach space \(E\). He applies the method of the Lie-Trotter product formula. Namely, if the solution of the Cauchy problem NEWLINE\[NEWLINE{\dot u}(t)=B u(t),\quad t\geq 0; \quad u(0)=fNEWLINE\]NEWLINE is given by a strongly continuous semigroup \(S(t)\) then the solution to \((*)\) is given (at least formally) by the Lie-Trotter product formula NEWLINE\[NEWLINEU(t)f=\lim_{n\to\infty} [T(\frac{t}{n}) S(\frac{t}{n})]^n f.\tag \(**\) NEWLINE\]NEWLINE In the first chapter the author constructs a semigroup \(U(t)\), generated by a sum of two generators \(A\) and \(B\), such that the Lie-Trotter formula does not hold. The second chapter is devoted to positive results in the linear case. The main is Theorem 2.2, where the usual range condition is replaced by the commutator condition NEWLINE\[NEWLINE\|T(t)S(t)f-S(t)T(t)f\|\leq t^\alpha M \|f\|,\quad \alpha>1, \quad 0\leq t\leq \delta.\tag \(***\) NEWLINE\]NEWLINE Applications to Ornstein-Uhlenbeck operators are also given. The nonlinear case is treated in chapter three. The commutator condition \((***)\) is replaced by the subcommutator condition: \(T(t)S(t)f-S(t)T(t)f \geq 0 \) on an ordered Banach space \(E\). In Theorem 3.15 it is proven that the nonlinear Lie-Trotter formula \((**)\) holds and moreover, \(S(t)T(t)f\leq U(t)f\leq T(t)S(t)f.\) In particular, studying the blow-up of the lower bound \(S(t)T(t)f\), the author deduces results about the blow-up of the solution \(U(t)f\). In the last chapter the author considers the case when \(Au=\sum a_{ij}(x)\frac{\partial^2 u}{\partial x_i \partial_j},\) \(Bu=\sum b_i(x)\frac{\partial u}{\partial x_i}, x\in {\mathbb R}^N. \) Since, in general, these operators are not generators, the above method of the Lie-Trotter formula fails. Instead, the maximum principle, \(L^p\) and Schauder estimates are used in order to prove that \(A+B\) is a generator. The regularity of the generated Feller semigroup is also investigated.