Abstract
In this paper, we are concerned with the asymptotic behavior of L∞ weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent dam** \( - \frac{m}{{{{(1 + t)}^\lambda }}}\). As \(\lambda \in (0,\tfrac{1}{7}]\), we prove that the L∞ weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation (GPME) in \({L^2}(\mathbb{R})\). As \(\lambda \in (\tfrac{1}{7},1)\), we prove that the L∞ weak-entropy solution converges to an expansion around the nonlinear diffusion wave in \({L^2}(\mathbb{R})\), which is the best asymptotic profile. The proof is based on intensive entropy analysis and an energy method.
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References
Chen G. Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics. III. Acta Math Sci, 1986, 6(1): 75–120
Chen S, Li H, Li J, Mei M, Zhang K. Global and blow-up solutions for compressible Euler equations with time-dependent dam**. J Differential Equations, 2020, 268: 5035–5077
Cui H -B, Yin H -Y, Zhang J -S, Zhu C -J. Convergence to nonlinear diffusion waves for solutions of Euler equations with time-depending dam**. J Differential Equations, 2018, 264: 4564–4602
Ding X, Chen G, Luo P. Convergence of the fractional step Lax-Friedrichs and Godunov scheme for isentropic system of gas dynamics. Commun Math Phys, 1989, 121: 63–84
Diperna R. Convergence of viscosity method for isentropic gas dynamics. Commun Math Phys, 1983, 91: 1–30
Geng S, Huang F. L1-convergence rates to the Barenblatt solution for the damped compressible Euler equations. J Differential Equations, 2019, 266(12): 7890–7908
Geng S, Huang F, Wu X. L1-convergence to generalized Barenblatt solution for compressible Euler equations with time-dependent dam**. SIAM J Math Anal, 2021, 53(5): 6048–6072
Geng S, Huang F, ** G, Wu X. The time asymptotic expansion for the compressible Euler equations with time-dependent dam**. Article MathSciNet Google Scholar
Hsiao L, Liu T. Nonlinear diffusive phenomena of nonlinear hyperbolic systems. Chin Ann Math, 1993, 14B: 465–480
Huang F, Marcati P, Pan R. Convergence to Barenblatt solution for the compressible Euler equations with dam** and vacuum. Arch Ration Mech Anal, 2005, 176: 1–24
Huang F, Pan R. Convergence rate for compressible Euler equations with dam** and vacuum. Arch Ration Mech Anal, 2003, 166: 359–376
Huang F, Pan R. Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum. J Differ Equ, 2006, 220: 207–233
Huang F, Pan R, Wang Z. L1 Convergence to the Barenblatt solution for compressible Euler equations with dam**. Arch Ration Mech Anal, 2011, 200: 665–689
Huang F, Wang Z. Convergence of viscosity solutions for isothermal gas dynamics. SIAM J Math Anal, 2002, 34(3): 595–610
Li H -T, Li J -Y, Mei M, Zhang K -J. Convergence to nonlinear diffusion waves for solutions of p-system with time-dependent dam**. J Math Anal Appl, 2017, 456: 849–871
Lions P L, Perthame B, Tadmor E. Kinetic formulation of the isentropic gas dynamics and p-systems. Commun Math Phys, 1994, 163: 169–172
Lions P L, Perthame B, Souganidis P. Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun Pure Appl Math, 1996, 49: 599–638
Liu T. Compressible flow with dam** and vacuum. Japan J Appl Math, 1996, 13: 25–32
Liu T, Yang T. Compressible Euler equations with vacuum. J Differ Equ, 1997, 140: 223–237
Liu T, Yang T. Compressible flow with vacuum and physical singularity. Methods Appl Anal, 2000, 7: 495–509
Mei M. Nonlinear diffusion waves for hyperbolic p-system with nonlinear dam**. J Differential Equations, 2009, 247: 1275–1296
Mei M. Best asymptotic profile for hyperbolic p-system with dam**. SIAM J Math Anal, 2010, 42: 1–23
Nishihara K. Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with dam**. J Differ Equ, 1996, 131: 171–188
Nishihara K. Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media. Proc Roy Soc Edinburgh Sect A, 2003, 133: 177–196
Nishihara K, Wang W, Yang T. Lp-convergence rate to nonlinear diffusion waves for p-system with dam**. J Differ Equ, 2000, 161: 191–218
Pan X H. Global existence of solutions to 1-d Euler equations with time-dependent dam**. Nonlinear Anal, 2016, 132: 327–336
Pan X H. Blow up of solutions to 1-d Euler equations with time-dependent dam**. J Math Anal Appl, 2016, 442: 435–445
Serre D, **ao L. Asymptotic behavior of large weak entropy solutions of the damped p-system. J Pure Differ Equ, 1997, 10: 355–368
Sugiyama Y. Singularity formation for the 1-D compressible Euler equations with variable dam** coefficient. Nonlinear Anal, 2018, 170: 70–87
Sugiyama Y. Remark on the global existence for the 1D compressible Euler equation with time-dependent dam**. (to appear)
Wirth J. Solution representations for a wave equation with weak dissipation. Math Methods Appl Sci, 2004, 27: 101–124
Wirth J. Wave equations with time-dependent dissipation. I. Non-effective dissipation. J Differential Equations, 2006, 222: 487–514
Wirth J. Wave equations with time-dependent dissipation. II. Effective dissipation. J Differential Equations, 2007, 232: 74–103
Zhao H. Convergence to strong nonlinear diffusion waves for solutions of p-system with dam**. J Differ Equ, 2001, 174: 200–236
Zheng Y. Global smooth solutions to the adiabatic gas dynamics system with dissipation terms. Chinese Ann Math, 1996, 17A: 155–162
Zhu C J. Convergence rates to nonlinear diffusion waves for weak entropy solutions to p-system with dam**. Sci China Ser A, 2003, 46: 562–575
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Dedicated to Professor Banghe LI on the Occasion of his 80th birthday
S. Geng’s research was supported in part by the National Natural Science Foundation of China (12071397) and Excellent Youth Project of Hunan Education Department (21B0165). F. Huang’s research was supported in part by the National Key R&D Program of China 2021YFA1000800 and the National Natural Science Foundation of China (12288201).
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Geng, S., Huang, F. & Wu, X. L2-convergence to nonlinear diffusion waves for Euler equations with time-dependent dam**. Acta Math Sci 42, 2505–2522 (2022). https://doi.org/10.1007/s10473-022-0618-6
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DOI: https://doi.org/10.1007/s10473-022-0618-6
Key words
- L 2-convergence
- compressible Euler Equations
- time asymptotic expansion
- time-dependent dam**
- relative entropy inequality