Abstract
In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.
Highlights
In this paper, we consider the first-order linear symmetric hyperbolic system with relaxation: Citation: Maekawa, Y.; Ueda, Y. nCharacterization of Dissipative ∂t u +Structures for First-Order Symmetric∑ A j ∂xj u + Lu = 0, t > 0, x ∈ Rn, j =1 u | t =0 = f, Hyperbolic System with General (1) n x∈R .Relaxation
Kawashima and Ueda [20] recently refined the entropy condition which can be applied to the compressible Euler–Maxwell system, where the key generalization is to allow the nonsymmetric relaxation
We study the linear system (1) for the case of general relaxation, and our goal is to provide the algebraic characterization in achieving the uniform dissipative estimate of order 1 without assuming
Summary
We consider the first-order linear symmetric hyperbolic system with relaxation: Citation: Maekawa, Y.; Ueda, Y. Is not enough to cover all physical models described by the balance laws, and, the theory still needs to be developed In this context, Kawashima and Ueda [20] recently refined the entropy condition which can be applied to the compressible Euler–Maxwell system, where the key generalization is to allow the nonsymmetric relaxation. We study the linear system (1) for the case of general relaxation, and our goal is to provide the algebraic characterization in achieving the uniform dissipative estimate of order 1 (see the definition in front of Theorem 1 below) without assuming (9). In Appendix A we recall the Gearhart-Prüss type theorem for the semigroup generated by the m-accretive operator on the Hilbert space, and in Appendix B we state the elementary fact about the nonnegative matrices with the spectral parameters on the imaginary axis These results are the key in our argument
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