In this paper, we study differential operators of the formfor matrices A, B ∈ C N ,N , where the eigenvalues of A have positive real parts. The sum A v(x) + Sx, ∇v(x) is known as the Ornstein-Uhlenbeck operator with an unbounded drift term defined by a skewsymmetric matrix S ∈ R d,d . Differential operators such as L ∞ arise as linearizations at rotating waves in time-dependent reaction-diffusion systems. The results of this paper serve as foundation for proving exponential decay of such waves. Under the assumption that A and B can be diagonalized simultaneously, we construct a heat kernel matrix H (x, ξ, t) of L ∞ that solves the evolution equation v t = L ∞ v. In the following, we study the Ornstein-Uhlenbeck semigroupin exponentially weighted function spaces. This is used to derive resolvent estimates for L ∞ in exponentially weighted L p -spaces L p θ (R d , C N ), 1 p < ∞, as well as in exponentially weighted C b -spaces C b,θ (R d , C N ).

In this paper we study perturbed Ornstein-Uhlenbeck operatorsfor simultaneously diagonalizable matrices A, B ∈ C N,N . The unbounded drift term is defined by a skew-symmetric matrix 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 D(Ap) of the generator Ap belonging to the Ornstein-Uhlenbeck semigroup coincides with the domain of L∞ in L p (R d , C N ) given byOne key assumption is a new L p -dissipativity condition |z| 2 Re w, Aw + (p − 2)Re w, z Re z, Aw γA|z| 2 |w| 2 ∀ z, w ∈ C N for some γA > 0. The proof utilizes the following ingredients. First we show the closedness of L∞ in L p and derive L p -resolvent estimates for L∞. Then we prove that the Schwartz space is a core of Ap and apply an L p -solvability result of the resolvent equation for Ap. A second characterization shows that the maximal domain even coincides with D p max (L0) = {v ∈ W 2,p | S·, ∇v ∈ L p }, 1 < p < ∞. This second characterization is based on the first one, and its proof requires L p -regularity for the Cauchy problem associated with Ap. Finally, we show a W 2,p -resolvent estimate for L∞ and an L p -estimate for the drift term S·, ∇v . Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.

Abstract. In this paper we study nonlinear problems for Ornstein-Uhlenbeck operatorswhere the matrix A ∈ R N,N is diagonalizable and has eigenvalues with positive real part, the map f : R N → R N is sufficiently smooth and the matrix S ∈ R d,d in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating waves in time-dependent reaction diffusion systems. We prove under appropriate conditions that every bounded classical solution v of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that v belongs to an exponentially weighted Sobolev space W 1,p. Several extensions of this basic result are presented: to complex-valued systems, to exponential decay in higher order Sobolev spaces and to pointwise estimates. We also prove that every bounded classical solution v of the eigenvalue problemdecays exponentially in space, provided Re λ lies to the right of the essential spectrum. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation. Our results form the basis for investigating nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains.

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Abstract. In this paper we study perturbed Ornstein-Uhlenbeck operatorsfor simultaneously diagonalizable matrices A, B ∈ C N,N . The unbounded drift term is defined by a skew-symmetric matrix S ∈ R d,d . Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. As shown in a companion paper, one key assumption to prove resolvent estimates ofwhich is a lower p-dependent bound of the first antieigenvalue of the diffusion matrix A. This relation provides a complete algebraic characterization and a geometric meaning of L pdissipativity for complex-valued Ornstein-Uhlenbeck operators in terms of the antieigenvalues of A. The proof is based on the method of Lagrange multipliers. We also discuss several special cases in which the first antieigenvalue can be given explicitly.