Well-posedness of hyperbolic systems with multiplicities and smooth coefficients

Abstract

We study hyperbolic systems with multiplicities and smooth coefficients. In the case of non-analytic, smooth coefficients, we prove well-posedness in any Gevrey class and when the coefficients are analytic, we prove C ∞ well-posedness. The proof is based on a transformation to block Sylvester form introduced by D'Ancona and Spagnolo (Boll UMI 8(1B):169-185, 1998) which increases the system size but does not change the eigenvalues. This reduction introduces lower order terms for which appropriate Levi-type conditions are found. These translate then into conditions on the original coefficient matrix. This paper can be considered as a generalisation of Garetto and Ruzhansky (Math Ann 357(2):401-440, 2013), where weakly hyperbolic higher order equations with lower order terms were considered.

On the wave equation with multiplicities and space-dependent irregular coefficients

“…Many authors have proven Gevrey wellposedness for hyperbolic Cauchy problems with multiplicities mainly when the coefficients depend only on time (see [CK, CK02, CDS, dAKi05, GR12, GR13, GR14b, KS] and references therein). The methods employed to obtain Gevrey well-posedness often combine algebraic transformations like symmetrisation or quasi-symmetrisation and energy estimates and require specific conditions of Levi-type on the lower order terms [dAS98, GR13,GJ17]. C ∞ well-posedness has also been obtained for some special classes of t-dependent equations and systems, for instance in [GR17,JT].…”

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

“…Note that since Γ x = Γ y = R n in Theorem 5.1, we may write the following assumption | det ∂ x ∂ y φ(x, y)| ≥ C > 0, for all x, y ∈ R n , instead of (24).…”

“…It is also not restrictive to assume that the matrix A is upper-triangular since conditions are given in [25] which allow the reduction into upper-triangular form of the system above. For a non exhausting overview on hyperbolic problems with multiplicities we refer the reader to [1,2,3,4,5,6,7,8,9,10,11,12,13,16,17,18,19,21,22,24,32,33,36,37,39].…”

“…More recently, advances in the theory of weakly hyperbolic systems with t-dependent coefficients have been obtained for systems of any size in presence of multiplicities with regular or low regular (Hölder) coefficients [16,22,23]. In addition, in [17] precise conditions on the lower order terms (Levi conditions) have been formulated to guarantee Gevrey and ultradistributional well-posedness. Previously very few results were known in the field for systems of a certain size (2 × 2, 3 × 3) [12,13] or of a certain form (for instance without lower order terms or with principal part of a certain form) [44].…”