Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping

Wu, Qiwei; Luan, Liping

doi:10.3934/cpaa.2021003

Cited by 7 publications

(2 citation statements)

References 33 publications

(43 reference statements)

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“…We remark that the Cauchy problem for the over-damped case (λ ∈ [−1, 0)), and under the condition of small perturbation of positive constant density, Ji and Mei [8] obtained the optimal decay estimates for solutions in R n (n ≥ 2). For more related results to the Euler and related equations, we refer readers to [1,5,7,9,13,14,15,19,20,23] and references therein.…”

Section: Introduction 1vacuum Free Boundary Problem and Related Resultsmentioning

confidence: 99%

Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Li,

Jia,

Zhang

et al. 2023

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We constructed a class of analytical, rotationaland self-similar solutions to the isentropic compressible Euler equations with time-dependent damping $\frac{\mu}{\left( 1+t\right) ^{\lambda}}%\rho\mathbf{u}$ ($\mu\geq0$, $\lambda\in\mathbb{R}$) and vacuum free boundary in cylindrical coordinates. These solutions capture the precise physical vacuum behavior that the sound speed is $C^{1/2}$-H\"older continuous across the boundary and are determined by the freeboundary $a(t)$, which is the solution of a second order nonlinear ordinarydifferential equation with parameters $\mu$ and $\lambda$ (see (1.12)) $). Furthermore, global well-posedness and asymptotic behavior of the free boundary $a(t)$ are presented in this paper. Precisely, we show that for $\lambda>1$ the freeboundary will grow linearly in time, and radial velocity $u^{r}$ and axial velocity $u^{z}$ are bounded, while the angular velocity $u^{\phi}$ and the radial derivatives of velocity components all tend to zero as $t\rightarrow+\infty$ (see Remark 1.4). However, if $\lambda\leq1$, the free boundary will grow sub-linearly in time. In particular, if $-1\leq \lambda\leq 1/(2\gamma-1)$, where $\gamma$ is the adiabatic exponent of polytropicgases, the free boundary will grow more slowly as $\lambda$ becomes smaller, and possesses a finite upper bound when $\lambda<-1$. Mathematics Subject Classifications (2020): 35R35, 76N10

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Section: Introduction 1vacuum Free Boundary Problem and Related Resultsmentioning

confidence: 99%

Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Li,

Jia,

Zhang

et al. 2023

Preprint

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“…Then, as in Wu and Luan, 34 we define

{\hat{n}}_i\left(x,t\right)&#x0003D;\left({C}_i&#x0002B;{\int}_0&#x0005E;t\left({J}_i&#x0005E;{-}\left(\tau \right)-{J}_i&#x0005E;{&#x0002B;}\left(\tau \right)\right) d\tau \right){m}_0(x),i&#x0003D;1,2,

{\hat{J}}_i\left(x,t\right)&#x0003D;{J}_i&#x0005E;{-}(t)&#x0002B;\left({J}_i&#x0005E;{&#x0002B;}(t)-{J}_i&#x0005E;{-}(t)\right){\int}_{-\infty}&#x0005E;x{m}_0(y) dy,i&#x0003D;1,2,

\hat{E}\left(x,t\right)&#x0003D;{E}&#x0005E;{&#x0002B;}(t){\int}_{-\infty}&#x0005E;x{m}_0(y) dy.

…”

Section: Preliminariesmentioning

confidence: 89%

Asymptotic stability of nonlinear diffusion waves for the bipolar quantum Euler–Poisson system with time‐dependent damping

Hou

2022

Math Methods in App Sciences

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We shall investigate the asymptotic behavior of solutions to the Cauchy problem for the one-dimensional bipolar quantum Euler-Poisson system with timedependent damping effects − J i (1+t) 𝜆 (i = 1, 2) for −1 < 𝜆 < 1. Applying the technical time-weighted energy method, we prove that the classical solutions to the Cauchy problem exist uniquely and globally, and time-algebraically converge to the nonlinear diffusion wave.

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Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space

2022

Journal of Mathematical Analysis and Applications

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Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping

Cited by 7 publications

References 33 publications

Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Asymptotic stability of nonlinear diffusion waves for the bipolar quantum Euler–Poisson system with time‐dependent damping

Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space

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