Analytical solutions to some one-dimensional compressible fluid equations

Jianwei Dong; Manwai Yuen; Litao Zhang; Junhui Zhu; Jianwei Dong; Manwai Yuen; Litao Zhang; Junhui Zhu

doi:10.3934/math.2026459

AIMS Mathematics

2026, Volume 11, Issue 4: 11154-11172. doi: 10.3934/math.2026459

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Analytical solutions to some one-dimensional compressible fluid equations

1.
School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450015, China
2.
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong, China

Received: 28 January 2026 Revised: 09 April 2026 Accepted: 10 April 2026 Published: 21 April 2026
MSC : 35C06, 35Q30, 35Q31

In this paper, we construct some self-similar analytical solutions for some one-dimensional compressible fluid equations. For the isentropic Euler equations with heat conduction, we present an analytical solution with the temperature decaying in time at the rate of $ \mathcal{O}((1+t)^{-1}) $ for the free boundary problem, provide an analytical solution for the Cauchy problem, and investigate its blowup and decay phenomena. We also construct some analytical solutions for the isothermal Euler equations with heat conduction. Moreover, we give an exact solution to the isentropic Euler equations with time-dependent damping and construct an analytical solution to the Navier-Stokes equations with density-dependent viscosity for $ \gamma = 3 $, respectively, where $ \gamma $ is the adiabatic exponent.
- Euler equations,
- Navier-Stokes equations,
- self-similar,
- analytical solutions
Citation: Jianwei Dong, Manwai Yuen, Litao Zhang, Junhui Zhu. Analytical solutions to some one-dimensional compressible fluid equations[J]. AIMS Mathematics, 2026, 11(4): 11154-11172. doi: 10.3934/math.2026459

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Abstract

In this paper, we construct some self-similar analytical solutions for some one-dimensional compressible fluid equations. For the isentropic Euler equations with heat conduction, we present an analytical solution with the temperature decaying in time at the rate of $ \mathcal{O}((1+t)^{-1}) $ for the free boundary problem, provide an analytical solution for the Cauchy problem, and investigate its blowup and decay phenomena. We also construct some analytical solutions for the isothermal Euler equations with heat conduction. Moreover, we give an exact solution to the isentropic Euler equations with time-dependent damping and construct an analytical solution to the Navier-Stokes equations with density-dependent viscosity for $ \gamma = 3 $, respectively, where $ \gamma $ is the adiabatic exponent.

References

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