In this paper we study perturbed Ornstein–Uhlenbeck operators $$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$ L ∞ v ( x ) = A ▵ v ( x ) + S x , ∇ v ( x ) - B v ( x ) , x ∈ R d , d ⩾ 2 , for simultaneously diagonalizable matrices $$A,B\in \mathbb {C}^{N,N}$$ A , B ∈ C N , N . The unbounded drift term is defined by a skew-symmetric matrix $$S\in \mathbb {R}^{d,d}$$ S ∈ 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 $$\mathcal {D}(A_p)$$ D ( A p ) of the generator $$A_p$$ A p belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of $$\mathcal {L}_{\infty }$$ L ∞ in $$L^p(\mathbb {R}^d,\mathbb {C}^N)$$ L p ( R d , C N ) given by $$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$ D loc p ( L 0 ) = v ∈ W loc 2 , p ∩ L p ∣ A ▵ v + S · , ∇ v ∈ L p , 1 < p < ∞ . One key assumption is a new $$L^p$$ L p -dissipativity condition $$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$ | z | 2 Re w , A w + ( p - 2 ) Re w , z Re z , A w ⩾ γ A | z | 2 | w | 2 ∀ z , w ∈ C N for some $$\gamma _A>0$$ γ A > 0 . The proof utilizes the following ingredients. First we show the closedness of $$\mathcal {L}_{\infty }$$ L ∞ in $$L^p$$ L p and derive $$L^p$$ L p -resolvent estimates for $$\mathcal {L}_{\infty }$$ L ∞ . Then we prove that the Schwartz space is a core of $$A_p$$ A p and apply an $$L^p$$ L p -solvability result of the resolvent equation for $$A_p$$ A p . In addition, we derive $$W^{1,p}$$ W 1 , p -resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.
Semigroup Forum – Springer Journals
Published: May 26, 2016
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