2011
DOI: 10.1080/03605302.2011.587492
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Non-Homogeneous Boundary Value Problems for Linear Dispersive Equations

Abstract: While the non-homogeneous boundary value problem for elliptic, hyperbolic and parabolic equations is relatively well understood, there are still few results for general dispersive equations. We define here a convenient class of equations comprising the Schrödinger equation, the Airy equation and linear "Boussinesq type" systems, which is in some sense a generalization of strictly hyperbolic equations, and for which we define a generalized Kreiss-Lopatinskiȋ condition. From the construction of generalized Kreis… Show more

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Cited by 15 publications
(29 citation statements)
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“…The estimate (16) below is also the direct analogue of the weighted in time estimates discussed in [BGS07, Chapter 4] and that are also invariant by the scaling (t, x) → (α t, α x) (the Laplace parameter γ below being then rescaled as γ → γ/α). As already mentioned, we plan to adapt the continuous dispersive estimates of [Aud12] to the framework of finite difference schemes and transparent boundary conditions in a near future. We now introduce the following terminology.…”
Section: Characterization Of Strong Stabilitymentioning
confidence: 99%
“…The estimate (16) below is also the direct analogue of the weighted in time estimates discussed in [BGS07, Chapter 4] and that are also invariant by the scaling (t, x) → (α t, α x) (the Laplace parameter γ below being then rescaled as γ → γ/α). As already mentioned, we plan to adapt the continuous dispersive estimates of [Aud12] to the framework of finite difference schemes and transparent boundary conditions in a near future. We now introduce the following terminology.…”
Section: Characterization Of Strong Stabilitymentioning
confidence: 99%
“…Since the Green-Naghdi model is nonlinear and dispersive, the analysis of non-trivial boundary conditions is rather challenging. Only few contributions on boundary conditions for the Green-Naghdi equations are available, while the articles [3,35] propose some analysis in a similar context. The fact that in many cases boundary conditions for numerical schemes are tailored to reproduce a specific phenomenon and justified only afterwards shows that we are still far away from a full understanding of boundary conditions for the Green-Naghdi and the water waves model.…”
Section: Introductionmentioning
confidence: 99%
“…The well-posedness of the IBVP (1.1) in a half line with solutions in the space C([0, T ]; H s (R + )) for s ≥ 0 is also discussed in [8] using boundary integral operator method (see also [24] for the recent study of the IBVP (1.1) in a half line). For higher dimensional cases, Audiard [2] investigated the non-homogeneous boundary value problem for some general linear dispersive equations and obtained a-priori estimates and well-posedness for the pure boundary value problems associated to such class of equations or the IBVPs of the linear Schrödinger equations in a half-space, while [3,4] study the corresponding IBVPs for the nonlinear Schrödinger equations in spatial domains that are the exterior of non-trapping compact or star-shaped obstacles. Here, we remark that the results obtained here are for the IBVPs of nonlinear Schrödinger equations in a half-space with optimal boundary data and the method used is the boundary integral operator method, which are totally different from those in [2,3,4].…”
Section: Introductionmentioning
confidence: 99%
“…In order to have the solution of (1.1) in the space C([0, T ]; H s (R×R + )) with s ≥ 0, while the initial the initial value ϕ(x, y) is chosen from H s (R × R + ), the optimal choice of the function space for the boundary data h(x, t) needs some discussion. Based upon the scaling argument, it seems natural (see [2,3]) to choose h from the space , where F[w] stands for the Fourier transform of w with respect to both x and t. The choose of h from H s (R 2 ) is optimal and the following definition of the well-posedness for the IBVP (1.1) is then natural. For small s ≥ 0, we need to address the meaning of solutions of the IBVP (1.1) satisfying the initial and boundary conditions.…”
Section: Introductionmentioning
confidence: 99%