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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 & 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. Google Scholar

[2]

J. ChungE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

R. Manas 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. Google Scholar

[12]

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

[13]

P. L. SachdevCh. 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. Google Scholar

[14]

P. L. SachdevK. 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. Google Scholar

[15]

P. L. SachdevK. 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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[19]

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

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. Google Scholar

[2]

J. ChungE. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

R. Manas 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. Google Scholar

[12]

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

[13]

P. L. SachdevCh. 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. Google Scholar

[14]

P. L. SachdevK. 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. Google Scholar

[15]

P. L. SachdevK. 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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[19]

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

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