6533b871fe1ef96bd12d10cb
RESEARCH PRODUCT
On the Cauchy problem for microlocally symmetrizable hyperbolic systems with log-Lipschitz coefficients
Ferruccio ColombiniDaniele Del SantoFrancesco FanelliGuy Métiviersubject
Pure mathematicsloss of derivativeshyperbolic equationGeneral MathematicsMathematics::Analysis of PDEsmicrolocal symmetrizabilityhyperbolic equations; hyperbolic systems; log-lipschitz coefficientsSpace (mathematics)01 natural sciencesMathematics - Analysis of PDEslog-Lipschitz regularity; loss of derivatives; global and local Cauchy problem; well-posedness; non-characteristic Cauchy problemwell-posednessFOS: MathematicsInitial value problem[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Uniqueness0101 mathematics[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]MathematicsSmoothness (probability theory)Spacetimelog-lipschitz coefficients010102 general mathematicsglobal and local Cauchy problemExtension (predicate logic)Lipschitz continuitynon-characteristic Cauchy problemhyperbolic equationshyperbolic systemMathematics Subject Classificationlog-Lipschitz regularityhyperbolic systemsAnalysis of PDEs (math.AP)description
International audience; The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in space problem we establish energy estimates with finite loss of derivatives, which is linearly increasing in time. This implies well-posedness in H ∞ , if the coefficients enjoy enough smoothness in x. From this result, by standard arguments (i.e. extension and convexification) we deduce also local existence and uniqueness. A huge part of the analysis is devoted to give an appropriate sense to the Cauchy problem, which is not evident a priori in our setting, due to the very low regularity of coefficients and solutions. 2010 Mathematics Subject Classification: 35L45 (primary); 35B45, 35B65 (secondary).
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2017-01-24 |