## Sylvie Benzoni-Gavage and Denis Serre

Print publication date: 2006

Print ISBN-13: 9780199211234

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199211234.001.0001

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# (p.443) A BASIC CALCULUS RESULTS

Source:
Multi-dimensional hyperbolic partial differential equations
Publisher:
Oxford University Press

The celebrated Gronwall Lemma is used repeatedly in this book. We state our most useful versions of it for convenience.

Lemma A.1 (Basic Gronwall Lemma) If u and f are smooth functions of t Є [0, T] such that

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with C0 ∈ R and C1 〉 0 then
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Proof The only trick in the proof is to show the final estimate for the righthand side

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of the original one. Since

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we easily get the inequality

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of which the claimed estimate is only a rougher version. □

A slightly more elaborate version that we often use is the following.

Lemma A.2 (Gronwall Lemma) If u and f are smooth functions of t Є [0, T] such that

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with C0 Є R and C1 〉 0 then

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(p.444) Lemma A.3 (‘Multidimensional’ Gronwall Lemma) Assume ℒ ⊂ R d+1 is a lens foliated by hypersurfaces Hθ and denote

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for θ Є [0, 1]. If u is a smooth function in the neighbourhood of L such that

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then there exists C depending only on C and L such that

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Proof The proof relies on the same trick as before but requires a little multidimensional calculus. Introducing parametric equations x = X(y, θ), t = T(y, θ) (y Є Ω ⊂ R d) for Hθ we have

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Hence

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Then we easily get the wanted estimate with

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Lemma A.4 (Discrete Gronwall Lemma) if a is a non-negative continuous function of s Є [0,t] and b is a non-decreasing function of s Є [0,t] such that

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for all (ε, s) with 0 〈 ε ≤ ε0 ∊ (0 t), s ∊ [0 t −ε], then
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(p.445) Proof The proof is fully elementary. Take D 〉 C and consider n Є N such that $ε n : = t n − 1$ satisfies

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For all s Є [0,t − εn], we have

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Therefore,

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We get the final estimate by letting n go to ∞ and then D go to C. □