The Lifespan of Classical Solutions to the (Damped) Compressible Euler Equations
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The Lifespan of Classical Solutions to the (Damped) Compressible Euler Equations Ka Luen Cheung1 · Sen Wong1 Received: 8 March 2020 / Revised: 26 September 2020 / Accepted: 6 October 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, the initial-boundary value problem of the original three-dimensional compressible Euler equations with (or without) time-dependent damping is considered. By considering a functional F(t, α, f ) weighted by a general time-dependent parameter function α and a general radius-dependent parameter function f , we show that if the initial value F|t=0 is sufficiently large, then the lifespan of the system is finite. Here, f can be any C 1 strictly increasing function such that the sum of initial values of f and α is non-negative. It follows that a class of conditions for non-existence of global classical solutions is established. Moreover, the conditions imply that a strong α will lead to a more unrestrained necessary condition for classical solutions of the system to exist globally in time. Keywords Blowup · Time-dependent damping · Compressible Euler equations · Global existence · Smooth solutions · Lifespan Mathematics Subject Classification 35B44 · 35L67 · 35Q31 · 35B30
1 Introduction and Main Result The three-dimensional compressible Euler equations with a time-dependent damping term can be expressed as
ρt + ∇ · (ρu) = 0, ρ [ut + (u · ∇)u] + ∇ p +
μ ρu = 0, (1 + t)λ
(1)
Communicated by Norhashidah Hj. Mohd. Ali.
B 1
Sen Wong [email protected] Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
123
K. L. Cheung , S. Wong
where ρ(t, x) : [0, T ) × R3 → [0, ∞) and u(t, x) : [0, T ) × R3 → R3 represent the density and the velocity of the fluid, respectively. Here, T > 0 is the lifespan of the C 1 solutions. Equation (1)1 comes from the mass conservation law, while (1)2 is a result of the momentum conservation law [6,7]. The p in (1)2 is the pressure function determined by a state equation. In this paper, we consider the state equation given by the adiabatic γ -law: p = K ργ ,
K > 0, γ ≥ 1.
(2)
The fluid is called isentropic and isothermal when γ > 1 and γ = 1, respectively. For other possibilities of the state equations, readers may refer to [5,10,11,16,17] and the references therein. μ The term (1+t) λ ρu with μ > 0 and λ ≥ 0 is the time-dependent term with damping μ coefficient (1+t)λ . When μ = 0, system (1) reduces to the original compressible Euler equations. In this case, the first finite lifespan (that is T < ∞) result was achieved by Sideris in [13], where the author considered the functional F(t) =
R3
x · ρudx
(3)
and showed that singularities will be developed on finite time based on the finite propagation speed property established for nonlinear hyperbolic equations in [15]. Subsequently, authors such as Zhu and Tu constructed different weighted functionals and established the corresponding fini
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