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.