Weakly hyperbolic systems with Hölder continuous coefficients

D’Ancona, Piero; Kinoshita, Tamotu; Spagnolo, Sergio

doi:10.1016/j.jde.2004.03.016

Cited by 6 publications

(13 citation statements)

References 7 publications

(6 reference statements)

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“…By Gronwall's Lemma on [c i , d i ] we get the inequality for |ξ| ≥ 1. This proves the C ∞ well-posedness of the Cauchy problem (1). Similarly, (30) implies the well-posedness of (1) in D ′ (R n ).…”

Section: 3supporting

confidence: 64%

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On $$C^\infty $$ C ∞ well-posedness of hyperbolic systems with multiplicities

Garetto

Ruzhansky

2017

Annali di Matematica

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In this paper, we study first-order hyperbolic systems of any order with multiple characteristics (weakly hyperbolic) and time-dependent analytic coefficients. The main question is when the Cauchy problem for such systems is well-posed in C ∞ and in D . We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantees the C ∞ well-posedness of the corresponding Cauchy problem.

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Section: 3supporting

confidence: 64%

“…The first results of this type for t-dependent hyperbolic systems of size 2×2 and 3×3 have been obtained by d'Ancona, Kinoshita and Spagnolo in [1,2]. For x-dependent 2×2 systems some results are also available, see e.g.…”

Section: Introductionmentioning

confidence: 99%

On $$C^\infty $$ C ∞ well-posedness of hyperbolic systems with multiplicities

Garetto

Ruzhansky

2017

Annali di Matematica

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“…Banach's fixed point theorem ensures the existence of a unique fixed point u 1 for the map G 0 1 . Hence, by assuming that the initial dataŨ 0 1 belongs to C([0, T * ], H s ) we conclude that there exists a unique u 1 ∈ C([0, T * ], H s ) solving (12). Note that the same argument proves that the operator I − G 0 1 is invertible on a sufficiently small interval in t since G 0 1 = I at t = 0.…”

Section: And With the Operators A(t X D X ) And B(t X D X ) Given Bymentioning

confidence: 67%

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

2018

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In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a wellposedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance k to it must be of order − k. This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing Communicated by Y. Giga.

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“…with ω(ε) ≥ cε r for some c, r > 0 and ϕ mollifier as in (9). The corresponding regularised operator…”

Section: Remark 22mentioning

confidence: 99%

On hyperbolic equations and systems with non-regular time dependent coefficients

Garetto

2015

Journal of Differential Equations

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In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than Hölder, namely bounded coefficients. As for second order equations in [14] we prove that such equations admit a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. The main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter. Classical solutions are recovered as limit of very weak solutions. Finally, by using a reduction to block Sylvester form we conclude that any first order hyperbolic system with non-regular coefficients is solvable in the very weak sense.

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Weakly hyperbolic systems with Hölder continuous coefficients

Cited by 6 publications

References 7 publications

On $$C^\infty $$ C ∞ well-posedness of hyperbolic systems with multiplicities

On $$C^\infty $$ C ∞ well-posedness of hyperbolic systems with multiplicities

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

On hyperbolic equations and systems with non-regular time dependent coefficients

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