Dispersive estimates for hyperbolic systems with time-dependent coefficients

Sugimoto

2015

Math. Ann.

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This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393-423, 2012), where dispersive equations were treated. For operators a(D x ) of order m satisfying the dispersiveness condition ∇a(ξ ) = 0 for ξ = 0, the global smoothing estimateis well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form

Section: This Is the Simplest Example Of Schrödinger Equations Couplementioning

confidence: 99%

“…In the case of dispersive and Strichartz estimates for higher order (in ∂ t ) equations the situation may be very delicate and in general depends on the rates of oscillations of c(t) (see e.g. Reissig [26] for the case of the time-dependent wave equation, or [8,31] for more general equations).…”

Section: Equations With Time-dependent Coefficientsmentioning

confidence: 99%

Smoothing estimates for non-dispersive equations

Sugimoto

2015

Math. Ann.

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“…In this case, even in the situation of the lower regularity of coefficients (C 1 ), one can analyse the global behaviour of solutions with respect to time (see [14]). The cases of constant coefficients and systems with controlled oscillations have been treated in [17,18], respectively.…”

Section: Theorem 1 ([13]) Assume That N = 1 and That The Differentialmentioning

confidence: 99%

Weakly hyperbolic equations with non-analytic coefficients and lower order terms

Garetto

2013

Math. Ann.

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In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under C k -regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the C ∞ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation (which have been untreatable until now in these problems) as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.

“…We note that the major advantage of the asymptotic integration method developed in [16] in comparison to other approaches, e.g. the diagonalisation schemes for systems as in [22], is the low C 1 regularity of coefficients in t sufficient for the construction compared to higher regularity required for other methods.…”

Section: Asymptotic Integrations For Linear Hyperbolic Systemmentioning

confidence: 99%

Global well-posedness of Kirchhoff systems

Matsuyama

Journal de Mathématiques Pures et Appliquées

2013

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The aim of this paper is to establish the $H^1$ global well-posedness for Kirchhoff systems. The new approach to the construction of solutions is based on the asymptotic integrations for strictly hyperbolic systems with time-dependent coefficients. These integrations play an important role to setting the subsequent fixed point argument. The existence of solutions for less regular data is discussed, and several examples and applications are presented.Comment: 24 page