An analysis to a model of tornado

Li, Tianhong

doi:10.1007/s00033-021-01647-y

Cited by 2 publications

(3 citation statements)

References 11 publications

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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%

“…Remark 1.7 For the solution (1.22) and (1.23) to the Euler equations, if we set β = 0, u z = η is a constant, which is similar to the case of Li in [13]. On the other hand, when β > 0, u z = η a β (t) r β is a function of r and t, it is different from the case of Yuen in [24] where u z is given by n ′ (t) n(t) x 3 , which has nothing to do with variable r (see (1.4)).…”

Section: Remark 16mentioning

confidence: 92%

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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%

Section: Remark 16mentioning

confidence: 92%

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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“…Tragedies caused by tsunamis in the last several decades highlighted that investigation of the physics of water waves are indispensable, and the obtained results can save human lives. One may fine studies on destructive weather phenomena one in [40].…”

Section: Introductionmentioning

confidence: 99%

Time-Dependent Analytic Solutions for Water Waves above Sea of Varying Depths

2022

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We investigate a hydrodynamic equation system which—with some approximation—is capable of describing the tsunami propagation in the open ocean with the time-dependent self-similar Ansatz. We found analytic solutions of how the wave height and velocity behave in time and space for constant and linear seabed functions. First, we study waves on open water, where the seabed can be considered relatively constant, sufficiently far from the shore. We found original shape functions for the ocean waves. In the second part of the study, we also consider a seabed which is oblique. Most of the solutions can be expressed with special functions. Finally, we apply the most common traveling wave Ansatz and present relative simple, although instructive solutions as well.

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An analysis to a model of tornado

Cited by 2 publications

References 11 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

Time-Dependent Analytic Solutions for Water Waves above Sea of Varying Depths

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