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January  2017, 16(1): 253-272. doi: 10.3934/cpaa.2017012

Higher order asymptotic for Burgers equation and Adhesion model

1. 

Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore-575 025, India

2. 

School of Mathematical Sciences, National Institute of Science Education and Research, Bhimpur-Padanpur, Via-Jatni, Khurda-752050, Odisha, India

3. 

Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore-575 025, India

Manas R. Sahoo, E-mail address: manas@niser.ac.in

Received  April 2016 Revised  September 2016 Published  November 2016

This paper is focused on the study of the large time asymptotic for solutions to the viscous Burgers equation and also to the adhesion model via heat equation. Using generalization of the truncated moment problem to a complex measure space, we construct asymptotic N-wave approximate solution to the heat equation subject to the initial data whose moments exist upto the order $2n+m$ and $i$-th order moment vanishes, for $i=0, 1, 2\dots m-1$. We provide a different proof for a theorem given by Duoandikoetxea and Zuazua [3], which plays a crucial role in error estimations. In addition to this we describe a simple way to construct an initial data in Schwartz class whose $m$ moments are equal to the $m$ moments of given initial data.

Citation: Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure and Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934/cpaa.2017012
References:
[1]

I. L. Chern and T. P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Commun. Math. Phys., 110 (1987), 503-517.

[2]

J. Chung, E. Kim and Y. J. Kim, Asymptotic agreement of moments and higher order contraction in the Burgers equation, J. Differential Equations, 248 (2010), 2417-2434. doi: 10.1016/j.jde.2010.01.006.

[3]

J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693-698.

[4]

S. N. Gurbatov and A. I. Saichev, New approximation in the adhesion model in the description of large scale structure of the universe, Cosmic velocity fields; Proc. 9th IAP Astrophysics, ed F. Bouchet and Marc Lachieze-Rey, (1993), 335-340.

[5]

E. Hopf, The Partial differential equation ut + uux = νuxx, Comm. Pure Appl. Math., 3 (1950), 201-230.

[6]

W. Jager and Y. G. Lu, On solutions to nonlinear reaction-diffusion -covection equations with degenerate diffusion, J. Differential Equations, 170 (2001), 1-21. doi: 10.1006/jdeq.2000.3800.

[7]

K. T. Joseph, One-dimensional adhesion model for large scale structures, Electron. J. Differential Equations, 2010 (2010), 1-15.

[8]

K. T. Joseph, A system of two conservation laws with flux conditions and small viscosity, J. Appl. Anal., 15 (2009), 247-267. doi: 10.1515/JAA.2009.247.

[9]

Y. J. Kim, A generalization of the moment problem to a complex measure space and an approximation technique using backward moments, Discrete Contin. Dyn. Syst., 30 (2011), 187-207. doi: 10.3934/dcds.2011.30.187.

[10]

J. C. Miller and A. J. Bernoff, Rates of convergence to self-similar solutions of Burgers equation, Stud. Appl. Math., 111 (2003), 29-40. doi: 10.1111/1467-9590.t01-2-00226.

[11]

Manas R. Sahoo, Generalized solution to a system of conservation laws which is not strictly hyperbolic, J. Math. Anal. Appl., 432 (2015), 214-232. doi: 10.1016/j.jmaa.2015.06.042.

[12]

J. Philip, Estimates of the age of a heat distribution, Ark. Mat., 7 (1968), 351-358.

[13]

P. L. Sachdev, Ch. Srinivasa Rao and K. T. Joseph, Analytic and numerical study of N-waves governed by the nonplanar Burgers equation, Stud. Appl. Math., 103 (1999), 89-120. doi: 10.1111/1467-9590.00122.

[14]

P. L. Sachdev, K. T. Joseph and K. R. C. Nair, Exact N-wave solutions for the non-planar Burgers equation, Proc. Roy. Soc. London Ser. A, 445 (1994), 501-517. doi: 10.1098/rspa.1994.0074.

[15]

P. L. Sachdev, K. T. Joseph and B. Mayil Vaganan, Exact N-wave solutions of generalized Burgers equations, Stud. Appl. Math., 97 (1996), 349-367. doi: 10.1002/sapm1996974349.

[16]

M. Oberguggenberger, Case study of a nonlinear, nonconservative, non-strictly hyperbolic system, Nonlinear Anal., 19 (1992), 53-79. doi: 10.1016/0362-546X(92)90030-I.

[17]

G. B. Whitham, Linear and Nonlinear Waves, John wiley and Sons, New York, 1974.

[18]

T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153-193. doi: 10.1111/1467-9590.00074.

[19]

T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation, Osaka J. Math., 44 (2007), 99-119.

show all references

References:
[1]

I. L. Chern and T. P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Commun. Math. Phys., 110 (1987), 503-517.

[2]

J. Chung, E. Kim and Y. J. Kim, Asymptotic agreement of moments and higher order contraction in the Burgers equation, J. Differential Equations, 248 (2010), 2417-2434. doi: 10.1016/j.jde.2010.01.006.

[3]

J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693-698.

[4]

S. N. Gurbatov and A. I. Saichev, New approximation in the adhesion model in the description of large scale structure of the universe, Cosmic velocity fields; Proc. 9th IAP Astrophysics, ed F. Bouchet and Marc Lachieze-Rey, (1993), 335-340.

[5]

E. Hopf, The Partial differential equation ut + uux = νuxx, Comm. Pure Appl. Math., 3 (1950), 201-230.

[6]

W. Jager and Y. G. Lu, On solutions to nonlinear reaction-diffusion -covection equations with degenerate diffusion, J. Differential Equations, 170 (2001), 1-21. doi: 10.1006/jdeq.2000.3800.

[7]

K. T. Joseph, One-dimensional adhesion model for large scale structures, Electron. J. Differential Equations, 2010 (2010), 1-15.

[8]

K. T. Joseph, A system of two conservation laws with flux conditions and small viscosity, J. Appl. Anal., 15 (2009), 247-267. doi: 10.1515/JAA.2009.247.

[9]

Y. J. Kim, A generalization of the moment problem to a complex measure space and an approximation technique using backward moments, Discrete Contin. Dyn. Syst., 30 (2011), 187-207. doi: 10.3934/dcds.2011.30.187.

[10]

J. C. Miller and A. J. Bernoff, Rates of convergence to self-similar solutions of Burgers equation, Stud. Appl. Math., 111 (2003), 29-40. doi: 10.1111/1467-9590.t01-2-00226.

[11]

Manas R. Sahoo, Generalized solution to a system of conservation laws which is not strictly hyperbolic, J. Math. Anal. Appl., 432 (2015), 214-232. doi: 10.1016/j.jmaa.2015.06.042.

[12]

J. Philip, Estimates of the age of a heat distribution, Ark. Mat., 7 (1968), 351-358.

[13]

P. L. Sachdev, Ch. Srinivasa Rao and K. T. Joseph, Analytic and numerical study of N-waves governed by the nonplanar Burgers equation, Stud. Appl. Math., 103 (1999), 89-120. doi: 10.1111/1467-9590.00122.

[14]

P. L. Sachdev, K. T. Joseph and K. R. C. Nair, Exact N-wave solutions for the non-planar Burgers equation, Proc. Roy. Soc. London Ser. A, 445 (1994), 501-517. doi: 10.1098/rspa.1994.0074.

[15]

P. L. Sachdev, K. T. Joseph and B. Mayil Vaganan, Exact N-wave solutions of generalized Burgers equations, Stud. Appl. Math., 97 (1996), 349-367. doi: 10.1002/sapm1996974349.

[16]

M. Oberguggenberger, Case study of a nonlinear, nonconservative, non-strictly hyperbolic system, Nonlinear Anal., 19 (1992), 53-79. doi: 10.1016/0362-546X(92)90030-I.

[17]

G. B. Whitham, Linear and Nonlinear Waves, John wiley and Sons, New York, 1974.

[18]

T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153-193. doi: 10.1111/1467-9590.00074.

[19]

T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation, Osaka J. Math., 44 (2007), 99-119.

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