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Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness
Claudia Garetto1, Christian Jäh1, Michael Ruzhansky2
11Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU UK.
Mathematische Annalen
|April 2, 2019
Summary
This study analyzes the well-posedness of hyperbolic systems with variable coefficients. It establishes conditions for solving the Cauchy problem, including systems with multiple characteristics and Jordan block structures.
Area of Science:
- Partial Differential Equations
- Mathematical Analysis
Background:
- The Cauchy problem for hyperbolic systems is fundamental in mathematical physics.
- Analyzing systems with space-time dependent coefficients and variable characteristics presents significant challenges.
Purpose of the Study:
- To establish well-posedness results for a general class of hyperbolic systems.
- To develop methods for transforming complex systems into a more manageable triangular form.
Main Methods:
- Utilizing anisotropic Sobolev spaces to analyze the well-posedness.
- Applying Schur-type triangularization techniques to reduce system complexity.
- Investigating conditions on lower-order terms and characteristic multiplicities.
Main Results:
- A well-posedness result is established for hyperbolic systems with upper triangular principal parts under specific conditions on lower-order terms.
- Conditions for Schur-type triangularization are provided for general systems, enabling the application of the primary result.
- Explicit constructions and examples are detailed for 2x2 and 3x3 systems.
Conclusions:
- The research provides a robust framework for analyzing the Cauchy problem in a broad class of hyperbolic systems.
- The developed methods offer practical approaches for handling systems with variable coefficients and complex characteristic structures.
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