Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems

Dynamics of Partial Differential Equations

2016

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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.

mentioning

confidence: 75%

Section: Assumptions and Main Resultsmentioning

confidence: 99%

Section: Assumptions and Main Resultsmentioning

confidence: 99%

“…Constant coefficient perturbations of complex Ornstein-Uhlenbeck operators. In the first step we review and collect results from [26,27,28,29] for the complex…”

Section: Far-field Linearization Consider the Nonlinear Problemmentioning

confidence: 99%

Section: Far-field Linearization Consider the Nonlinear Problemmentioning

confidence: 99%

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The identification problem for complex-valued Ornstein–Uhlenbeck operators in $$L^p(\mathbb {R}^d,\mathbb {C}^N)$$ L p ( R d , C N )

2016

Semigroup Forum

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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.