Stable pencils of hyperbolic polynomials and the Cauchy problem for hyperbolic equations with a small parameter at the highest derivatives

Abstract

Abstract. We study pencils of hyperbolic polynomials of the form R (τ, ξ, where P j (τ, ξ) is a real homogeneous polynomials of degree m − j resolved with respect to the highest power of τ and P j (1, 0) = 1; the numbers γ 0 , . . . , γ N are positive. In the first part of the paper we find necessary and close to sufficient conditions of stability of the polynomial R(τ, ξ) (i.e., the condition that its roots τ j (ξ) lie in the open upper half-plane of the complex plane). This problem is closely related to the problem on uniform (with respect to a small parameter) estimates for the solution of the Cauchy problem for hyperbolic equations with a small parameter. The latter problem (both for constant and variable coefficients) is the topic of the second part of the paper.

On the Properties of the Representation of the Boltzmann Kinetic Equation in the Basis of Hermite Functions

“…is positive on the cone K 0,3;1,4 for arbitrary [6]. In the first case, we obtain inequality (23), and in the second case, we arrive at (24).…”

Global time estimates for solutions to equations of dissipative type

“…The size of these systems depends on the number of moments and dimension of the space. Some examples of these systems and their stability has been analysed in [VR03].…”

“…In [VR04] it was shown that P M +1 (τ, ξ) ≡ 0, γ M P M (τ, ξ) = M!τ for some γ M > 0, and γ M −1 P M −1 (τ, ξ) = M! M +1 k=2 1 k−1 τ 2 − M!|ξ| 2 for some γ M −1 > 0.…”

Dispersive and Strichartz Estimates for Hyperbolic Equations with Constant Coefficients

“…Clearly, in this situation it is hard to find the roots explicitly, and, therefore, we need some procedure of determining what are the general properties of the characteristics roots, and how to derive the time decay rate from these properties. Thus, in [Rad03] and [VR04] it is discussed when such polynomials are stable. In this case, the analysis of this paper will guarantee the decay rate, e.g.…”

“…Indeed, making the decomposition in the space of velocities into the Hermite basis, and writing equations for the space-time coefficients produces a hyperbolic system for infinitely many coefficients (see e.g. [VR03], [VR04], [ZR04], and Section 8.5). The Galerkin approximation of this system leads to a family of systems with sizes increasing to infinity.…”