Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains

Dong, Jianwei; Yuen, Man Wai

doi:10.1007/s00033-020-01392-8

Cited by 4 publications

(3 citation statements)

References 36 publications

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“…For λ > 0, (1.1) corresponds to the time-dependent damping case. On some fixed bounded domains and for some special initial data, Dong and Yuen in [3] studied the blowup of radial solutions to the compressible Euler equations with or without damping, by introducing some new averaged quantities. While for the free boundary problem, Dong and Li in [4] constructed a class of spherically symmetric and self-similar analytical solutions in R 3 when λ > 1, where the free boundary tends to +∞ at an algebraic rate not more than C(1 + t) 2 as t → +∞.…”

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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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

View full text Add to dashboard Cite

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“…Condition ( 3) is different in its nature: it controls the behaviour of solution as |x| → ∞ and do not prescribes any additional requirement on ρ. In fact, (3) can be applied even to solutions with infinite mass.…”

Section: Remarksmentioning

confidence: 99%

“…The present work seems to be a first attempt to use compactly supported moment function and describe the properties of the smooth solution of the full compressible Euler equations inside a fixed space domain. In addition, we use this new kind of moment function to find sufficient conditions for the behavior of the formation of a solution singularity and to determine the necessary behavior of a globally smooth solution in time as |x| → ∞ (see [3] for another kind of compactly supported moment functions applied to a simpler situation).…”

Section: Introductionmentioning

confidence: 99%

Localization of the formation of singularities in multidimensional compressible Euler equations

Rozanova

2020

Preprint

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We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial stationary state. We prove the blowup results using the characteristics of the propagation of the solution in space and find upper and lower bounds for the density of a smooth solution in a given region of space in terms of the initial data. To solve the problems, we introduce a special family of integral functionals and study their temporal dynamics.

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Analytical solutions to the compressible Euler equations with time-dependent damping and free boundaries

Dong

2022

Journal of Mathematical Physics

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In this paper, we study a class of analytical solutions to the compressible Euler equations with time-dependent damping [Formula: see text], which describe compressible fluids moving into outer vacuum. Under the continuous density condition across the free boundaries separating the fluid from vacuum, we construct a class of spherically symmetric and self-similar analytical solutions in [Formula: see text]. The global-in-time existence of such solutions is proved for μ > 0 and λ > 1. Moreover, the free boundary tends to + ∞ at an algebraic rate not more than C(1 + t)2 as t → + ∞.

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Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains

Cited by 4 publications

References 36 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

Localization of the formation of singularities in multidimensional compressible Euler equations

Analytical solutions to the compressible Euler equations with time-dependent damping and free boundaries

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