Decay estimates for hyperbolic systems

Uniqueness of solutions of genuinely nonlinear n × n strictly hyperbolic systems of balance laws is established moving from Oleïnik-type decay estimates. As middle step, the result relies on the fulfillment of a condition which controls the local oscillation of the solution in a forward neighborhood of each point in the t–x plane.

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Decay Estimates of Solutions for Quasi-Linear Hyperbolic Systems of Viscoelasticity

SIAM Journal on Mathematical Analysis ◽

10.1137/11083900x ◽

2012 ◽

Vol 44 (3) ◽

pp. 1976-2001 ◽

Cited By ~ 6

Author(s):

Priyanjana M. N. Dharmawardane ◽

Tohru Nakamura ◽

Shuichi Kawashima

Keyword(s):

Hyperbolic Systems ◽

Decay Estimates ◽

Estimates Of Solutions

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The optimal decay estimates on the framework of Besov spaces for the Euler–Poisson two-fluid system

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202515500463 ◽

2015 ◽

Vol 25 (10) ◽

pp. 1813-1844 ◽

Cited By ~ 1

Author(s):

Jiang Xu ◽

Shuichi Kawashima

Keyword(s):

Besov Spaces ◽

Hyperbolic Systems ◽

Low Frequency ◽

Direct Consequence ◽

Decay Rates ◽

Decay Estimates ◽

Fluid System ◽

Optimal Decay Rates ◽

Optimal Decay ◽

Two Fluid

In this paper, we are concerned with the optimal decay estimates for the Euler–Poisson two-fluid system. It is first revealed that the irrotationality of the coupled electronic field plays a key role such that the two-fluid system has the same dissipative structure as generally hyperbolic systems satisfying the Shizuta–Kawashima condition. This fact inspires us to obtain decay properties for linearized systems in the framework of Besov spaces. Furthermore, various decay estimates of solution and its derivatives of fractional order are deduced by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As the direct consequence, the optimal decay rates of Lp(ℝ3)-L2 (ℝ3) (1 ≤ p < 2) type for the Euler–Poisson two-fluid system are also shown. Compared with previous works in Sobolev spaces, a new observation is that the difference of variables exactly consists of a one-fluid Euler–Poisson equations, which leads to the sharp decay estimates for velocities.

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Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2019073 ◽

2019 ◽

Vol 39 (4) ◽

pp. 1651-1684

Author(s):

Thinh Tien Nguyen ◽

Keyword(s):

Hyperbolic Systems ◽

Asymptotic Limit ◽

Decay Estimates

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$$L^p$$Lp-$$L^q$$Lq Decay Estimates for Dissipative Linear Hyperbolic Systems in 1D

Theory, Numerics and Applications of Hyperbolic Problems II - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-3-319-91548-7_24 ◽

2018 ◽

pp. 305-319

Author(s):

Corrado Mascia ◽

Thinh Tien Nguyen

Keyword(s):

Hyperbolic Systems ◽

Decay Estimates

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DECAY RATES FOR SEMILINEAR VISCOELASTIC SYSTEMS IN WEIGHTED SPACES

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891612500026 ◽

2012 ◽

Vol 09 (01) ◽

pp. 67-103 ◽

Cited By ~ 2

Author(s):

REINHARD RACKE ◽

BELKACEM SAID-HOUARI

Keyword(s):

Linear Problem ◽

Hyperbolic Systems ◽

Low Frequency ◽

Pointwise Estimate ◽

Decay Rates ◽

Decay Estimates ◽

Weighted Energy Method ◽

Optimal Decay ◽

The Fourier Transform ◽

Semilinear Problem

We consider a class of second-order hyperbolic systems which describe viscoelastic materials, and we extend the recent results by Dharmawardane and Conti et al. More precisely, for all initial data (u0, u1)∈(Hs+1(ℝN) ∩ L1, γ(ℝN))×(Hs(ℝN) ∩ L1, γ(ℝN)) with γ∈[0, 1], we derive faster decay estimates for both dissipative structure or regularity-loss type models. To this end, we first transform our problem into Fourier space and then, by using the pointwise estimate derived by Dharmawardane et al., combined with a device to treat the Fourier transform in the low frequency region, we derive optimal decay results for the solutions to our problem. Finally, we use these decay estimates for the linear problem, combined with the weighted energy method introduced by Todorova and Yordanov, and tackle a semilinear problem.

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