American Institute of Mathematical Sciences

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December  2012, 5(4): 857-872. doi: 10.3934/krm.2012.5.857

Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities

Masashi Ohnawa 1,

1. 

Research Institute of Nonlinear Partial Differential Equations, Organization for University Research Initiatives, Waseda University, 3-4-1 Ohkubo Shinjuku, Tokyo 169-8555, Japan

Received  April 2012 Revised  August 2012 Published  November 2012

The present paper is concerned with the asymptotic behavior of a discontinuous solution to a model system of radiating gas. As we assume that an initial data has a discontinuity only at one point, so does the solution. Here the discontinuous solution is supposed to satisfy an entropy condition in the sense of Kruzkov. Previous researches have shown that the solution converges uniformly to a traveling wave if an initial perturbation is integrable and is small in the suitable Sobolev space. If its anti-derivative is also integrable, the convergence rate is known to be $(1+t)^{-1/4}$ as time $t$ tends to infinity. In the present paper, we improve the previous result and show that the convergence rate is exactly the same as the spatial decay rate of the initial perturbation.
Keywords: asymptotic stability, convergence rate, shock wave, weighted energy method., Hyperbolic-elliptic coupled system.
Mathematics Subject Classification: Primary: 35B35, 35B40, 35B45, 35M10; Secondary: 76N1.
Citation: Masashi Ohnawa. Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities. Kinetic & Related Models, 2012, 5 (4) : 857-872. doi: 10.3934/krm.2012.5.857
References:
[1]

R. Duan, K. Fellner and C. Zhu, Energy method for multi-dimensional balance laws with non-local dissipation, J. Math. Pures Appl., 93 (2010), 572-598. doi: 10.1016/j.matpur.2009.10.007.  Google Scholar

[2]

W. Gao, L. Ruan and C. Zhu, Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$ dimensions J. Differential Equations, 244 (2008), 2614-2640. doi: 10.1016/j.jde.2008.02.023.  Google Scholar

[3]

K. Hamer, Nonlinear effects on the propagation of sounds waves in a radiating gas, Quarter J. Mech. Appl. Math., 24 (1971), 155-168. doi: 10.1093/qjmam/24.2.155.  Google Scholar

[4]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.  Google Scholar

[5]

S. Kawashima and S. Nishibata, Weak solutions with a shock to a model system of the radiating gas, Sci. Bull. Josai Univ. special issue, 5 (1998), 119-130.  Google Scholar

[6]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117. doi: 10.1137/S0036141097322169.  Google Scholar

[7]

S. Kawashima and S. Nishibata, Cauchy problem for a model system of the radiating gas: Weak solutions with a jump and classical solutions, Math. Models. Meth. Sci., 9 (1999), 69-91.  Google Scholar

[8]

S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500.  Google Scholar

[9]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of radiating gas, Kyushu J. Math., 58 (2004), 211-250. doi: 10.2206/kyushujm.58.211.  Google Scholar

[10]

S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sbor., 10 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.  Google Scholar

[11]

C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009), 2165-2206. doi: 10.1137/09076026X.  Google Scholar

[12]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739.  Google Scholar

[13]

S. Nishibata, Asymptotic behavior of solutions to a model system of radiating gas with discontinuous initial data, Math. Models. Meth. Appl. Sci., 8 (2000), 1209-1231.  Google Scholar

[14]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcialaj Ekvacioj, 41 (1998), 107-132.  Google Scholar

[15]

M. Nishikawa and S. Nishibata, Convergence rates toward the traveling waves for a model system of the radiating gas, Math. Meth. Appl. Sci., 30 (2007), 649-663. doi: 10.1002/mma.800.  Google Scholar

[16]

D. Serre, $L^1$-stability of constants in a model for radiating gases, Comm. Math. Sci., 1 (2003), 197-205.  Google Scholar

[17]

M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinetic Related Models, 4 (2011), 569-588.  Google Scholar

[18]

W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions, Nonlinear Anal., 71 (2009), 1180-1195. doi: 10.1016/j.na.2008.11.050.  Google Scholar

show all references

References:
[1]

R. Duan, K. Fellner and C. Zhu, Energy method for multi-dimensional balance laws with non-local dissipation, J. Math. Pures Appl., 93 (2010), 572-598. doi: 10.1016/j.matpur.2009.10.007.  Google Scholar

[2]

W. Gao, L. Ruan and C. Zhu, Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$ dimensions J. Differential Equations, 244 (2008), 2614-2640. doi: 10.1016/j.jde.2008.02.023.  Google Scholar

[3]

K. Hamer, Nonlinear effects on the propagation of sounds waves in a radiating gas, Quarter J. Mech. Appl. Math., 24 (1971), 155-168. doi: 10.1093/qjmam/24.2.155.  Google Scholar

[4]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.  Google Scholar

[5]

S. Kawashima and S. Nishibata, Weak solutions with a shock to a model system of the radiating gas, Sci. Bull. Josai Univ. special issue, 5 (1998), 119-130.  Google Scholar

[6]

S. Kawashima and S. Nishibata, Shock waves for a model system of the radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117. doi: 10.1137/S0036141097322169.  Google Scholar

[7]

S. Kawashima and S. Nishibata, Cauchy problem for a model system of the radiating gas: Weak solutions with a jump and classical solutions, Math. Models. Meth. Sci., 9 (1999), 69-91.  Google Scholar

[8]

S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500.  Google Scholar

[9]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of radiating gas, Kyushu J. Math., 58 (2004), 211-250. doi: 10.2206/kyushujm.58.211.  Google Scholar

[10]

S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sbor., 10 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.  Google Scholar

[11]

C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009), 2165-2206. doi: 10.1137/09076026X.  Google Scholar

[12]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96. doi: 10.1007/BF02099739.  Google Scholar

[13]

S. Nishibata, Asymptotic behavior of solutions to a model system of radiating gas with discontinuous initial data, Math. Models. Meth. Appl. Sci., 8 (2000), 1209-1231.  Google Scholar

[14]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcialaj Ekvacioj, 41 (1998), 107-132.  Google Scholar

[15]

M. Nishikawa and S. Nishibata, Convergence rates toward the traveling waves for a model system of the radiating gas, Math. Meth. Appl. Sci., 30 (2007), 649-663. doi: 10.1002/mma.800.  Google Scholar

[16]

D. Serre, $L^1$-stability of constants in a model for radiating gases, Comm. Math. Sci., 1 (2003), 197-205.  Google Scholar

[17]

M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinetic Related Models, 4 (2011), 569-588.  Google Scholar

[18]

W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions, Nonlinear Anal., 71 (2009), 1180-1195. doi: 10.1016/j.na.2008.11.050.  Google Scholar

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