We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar conservation laws, nonlinear transport equations, Hamilton-Jacobi equations and fully nonlinear elliptic equations. Motivated by the application in nonlinear transport equations, we also prove basic results on flows generated by Lipschitz almost periodic vector fields, which are of interest in their own. In our analysis, an important role is played by the so-called Bohr compactification G(N) of R-N; this is a natural parameter space for the Young measures. Our homogenization results provide also the asymptotic behavior for the whole set of G(N)-translates of the solutions, which is in the spirit of recent studies on the homogenization of stationary ergodic processes.
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Grant MacEwan Univ, Dept Math Sci, Edmonton, AB T5J 4S2, Canada
Inst Math Simon Stoilow, Bucharest, RomaniaGrant MacEwan Univ, Dept Math Sci, Edmonton, AB T5J 4S2, Canada
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Catholic Kwandong Univ, Dept Math Educ, Kangnung 25601, Gangwon, South Korea
KIAS, 85 Hoegiro, Seoul 02455, South KoreaCatholic Kwandong Univ, Dept Math Educ, Kangnung 25601, Gangwon, South Korea
Lee, Jeong-Yup
Lenz, Daniel
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Friedrich Schiller Univ Jena, Fak Math & Informat, D-07737 Jena, GermanyCatholic Kwandong Univ, Dept Math Educ, Kangnung 25601, Gangwon, South Korea
Lenz, Daniel
Richard, Christoph
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Friedrich Alexander Univ Erlangen Nurnberg, Dept Math, Cauerstr 11, D-91058 Erlangen, GermanyCatholic Kwandong Univ, Dept Math Educ, Kangnung 25601, Gangwon, South Korea
Richard, Christoph
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Sing, Bernd
Strungaru, Nicolae
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MacEwan Univ, Dept Math Sci, 10700-104 Ave, Edmonton, AB T5J 4S2, Canada
Inst Math Simon Stoilow, Bucharest, RomaniaCatholic Kwandong Univ, Dept Math Educ, Kangnung 25601, Gangwon, South Korea
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Mathematics Institute, University of Warwick, Coventry,CV4 7AL, United KingdomMathematics Institute, University of Warwick, Coventry,CV4 7AL, United Kingdom
Arroyo-Rabasa, Adolfo
Diermeier, Johannes
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Institute of Applied Mathematics, University of Bonn, Bonn,D-53115, GermanyMathematics Institute, University of Warwick, Coventry,CV4 7AL, United Kingdom