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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sbor., 10 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.
[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.
[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.
[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.
[14]	M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcialaj Ekvacioj, 41 (1998), 107-132.
[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.
[16]	D. Serre, $L^1$-stability of constants in a model for radiating gases, Comm. Math. Sci., 1 (2003), 197-205.
[17]	M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinetic Related Models, 4 (2011), 569-588.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR Sbor., 10 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.
[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.
[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.
[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.
[14]	M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcialaj Ekvacioj, 41 (1998), 107-132.
[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.
[16]	D. Serre, $L^1$-stability of constants in a model for radiating gases, Comm. Math. Sci., 1 (2003), 197-205.
[17]	M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics, Kinetic Related Models, 4 (2011), 569-588.
[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.

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