Abstract
We consider the Cauchy problem for the multidimensional Burgers equation with a small dissipation parameter and use the matching method to construct an asymptotic solution near the singularity determined by the vector field structure at the initial instant. The method that we use allows tracing the evolution of the solution with a hierarchy of differently scaled structures and giving a rigorous mathematical definition of the asymptotic solution in the leading approximation. We discuss the relation of the considered problem to different models in fundamental and applied physics.
Similar content being viewed by others
References
G. B. Whitham, Linear and Non-Linear Waves, Wiley, New York (1974).
O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics [in Russian], Nauka, Moscow (1975); English transl., Consultants Bureau, New York (1977).
S. N. Gurbatov, O. V. Rudenko, and A. I. Saichev, Waves and Structures in Nonlinear Nondispersive Media: Applications to Nonlinear Acoustics [in Russian], Fizmatlit, Moscow (2008); English transl.: Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics, Springer, Berlin (2011).
F. Bernardeau and P. Valageas, “Merging and fragmentation in the Burgers dynamics,” Phys. Rev. E, 82, 016311 (2010); arXiv:0912.3603v2 [astro-ph.CO] (2009).
S. N. Gurbatov, A. I. Saichev, and S. F. Shandarin, “Large-scale structure of the Universe: The Zeldovich approximation and the adhesion model,” Phys. Usp., 55, 223–249 (2012); arXiv:astro-ph/9311075v1 (1993).
A. L. Melott, S. F. Shandarin, and D. H. Weinberg, “A test of the adhesion approximation for gravitational clustering,” Astrophys. J., 428, 28–34 (1994); arXiv:astro-ph/9311075v1 (1993).
V. I. Arnol’d, Singularities of Caustics and Wave Fronts [in Russian], Fazis, Moscow (1996); English transl. prev. ed. (Math. Its Appl. Sov. Ser., Vol. 62), Kluwer, Dordrecht (1990).
I. A. Bogaevsky, “Reconstructions of singularities of minimum functions, and bifurcations of shock waves of the Burgers equation with vanishing viscosity,” Leningrad Math. J., 1, 807–823 (1990).
I. A. Bogaevsky, “Discontinuous gradient differential equations and trajectories in calculus of variations,” Sb. Math., 197, 1723–1751 (2006).
A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems [in Russian], Nauka, Moscow (1989); English transl., Amer. Math. Soc., Providence, R. I. (1991).
S. V. Zakharov, “Singular asymptotics in the Cauchy problem for a parabolic equation with a small parameter [in Russian],” Tr. Inst. Mat. Mekh., 21, 97–104 (2015).
S. V. Zakharov, “Singularities of A and B types in asymptotic analysis of solutions of a parabolic equation,” Funct. Anal. Appl., 49, 307–310 (2015).
S. V. Zakharov, “The Cauchy problem for a quasilinear parabolic equation with a large initial gradient and low viscosity,” Comput. Math. Math. Phys., 50, 665–672 (2010).
T. Hollowood, J. L. Miramontes, and J. S. Guillén, “Generalized integrability and two-dimensional gravitation,” Theor. Math. Phys., 95, 552–567 (1993).
J. Schnittger and U. Ellwanger, “Nonperturbative conditions for local Weyl invariance on a curved world sheet,” Theor. Math. Phys., 95, 643–662 (1993).
S. Kachru, X. Liu, and M. Mulligan, “Gravity duals of Lifshitz-like fixed points,” Phys. Rev. D, 78, 106005 (2008); arXiv:0808.1725v2 [hep-th] (2008).
I. Arav, S. Chapman, and Y. Oz, “Lifshitz scale anomalies,” JHEP, 1502, 078 (2015); arXiv:1410.5831v1 [hep-th] (2014).
S. V. Zakharov, “The Cauchy problem for a quasilinear parabolic equation with two small parameters,” Dokl. Math., 78, 769–770.
S. V. Zakharov, “Asymptotic calculation of the heat distribution on a plane,” Proc. Steklov Inst. Math. (Suppl. 1), 296, 243–249 (2017).
K. Stewartson, “On almost rigid rotations: Part 2,” J. Fluid Mech., 26, 131–144 (1966).
F. H. Busse, “On Howard’s upper bound for heat transport by turbulent convection,” J. Fluid Mech., 37, 457–477 (1969).
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the complex Program for Basic Research, Ural Branch, RAS, “Analytic, asymptotic, and numerical methods for constructing direct, inverse, and singularly perturbed problems of mathematical physics” (Project No. 0387-2016-0039).
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 196, No. 1, pp. 42–49, July, 2018.
Rights and permissions
About this article
Cite this article
Zakharov, S.V. Asymptotic Solution of the Multidimensional Burgers Equation Near A Singularity. Theor Math Phys 196, 976–982 (2018). https://doi.org/10.1134/S0040577918070048
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577918070048