# Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt B3

## Numerische Analyse äquivarianter Evolutionsgleichungen

14-067 Denny Otten.
The Identification Problem for complex-valued Ornstein-Uhlenbeck Operators in $$L^p(\mathbb{R}^d,\mathbb{C}^N)$$

In this paper we study perturbed Ornstein-Uhlenbeck operators $\left[\mathcal{L}_{\infty} v\right](x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle-B v(x),\,x\in\mathbb{R}^d,\,d\geq 2,$ for simultaneously diagonalizable matrices $$A,B\in\mathbb{C}^{N,N}$$. The unbounded drift term is defined by a skew-symmetric matrix $$S\in\mathbb{R}^{d,d}$$. Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain $$\mathcal{D}(A_p)$$ of the generator $$A_p$$ belonging to the Ornstein-Uhlenbeck semigroup coincides with the domain of $$\mathcal{L}_{\infty}$$ in $$L^p(\mathbb{R}^d,\mathbb{C}^N)$$ given by $\mathcal{D}^p_{\mathrm{loc}}(\mathcal{L}_0)=\left\{v\in W^{2,p}_{\mathrm{loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot,\nabla v\right\rangle\in L^p\right\},\,1\lt p \lt\infty.$ One key assumption is a new $$L^p$$-dissipativity condition $|z|^2\text{Re}\left\langle w,Aw\right\rangle + (p-2)\text{Re}\left\langle w,z\right\rangle\text{Re}\left\langle z,Aw\right\rangle\geq \gamma_A |z|^2|w|^2\;\forall\,z,w\in\mathbb{C}^N$ for some $$\gamma_A>0$$. The proof utilizes the following ingredients. First we show the closedness of $$\mathcal{L}_{\infty}$$ in $$L^p$$ and derive $$L^p$$-resolvent estimates for $$\mathcal{L}_{\infty}$$. Then we prove that the Schwartz space is a core of $$A_p$$ and apply an $$L^p$$-solvability result of the resolvent equation for $$A_p$$. A second characterization shows that the maximal domain even coincides with $\mathcal{D}^p_{\mathrm{max}}(\mathcal{L}_0)=\{v\in W^{2,p}\mid \left\langle S\cdot,\nabla v\right\rangle\in L^p\},\,1 \lt p \lt\infty.$ This second characterization is based on the first one, and its proof requires $$L^p$$-regularity for the Cauchy problem associated with $$A_p$$. Finally, we show a $$W^{2,p}$$-resolvent estimate for $$\mathcal{L}_{\infty}$$ and an $$L^p$$-estimate for the drift term $$\left\langle S\cdot,\nabla v\right\rangle$$. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.