Abstract
This paper is concerned with the characterization of the weak limits (delta waves)\(\mathop {\lim }\limits_{\varepsilon \to 0} u^\varepsilon \) associated to the Cauchy problem for the Burgers' equation and the inviscid Burgers' equation with strongly singular initial data in the form of a regularization by smooth mollifiers of sums of derivatives of Dirac measures. By means of Laplace's method we give the precise asymptotic expansion of the solutionsu ɛ in powers of\(\sqrt \varepsilon \). Then we apply these asymptotics in order to classify completely all possible delta waves under a suitable nondegeneracy condition on some mollifiers regularizing the leading singular term of the initial data. We propose also certain stability results for the weak limits under suitable perturbations of the initial data.
Similar content being viewed by others
References
Biagioni H, Oberguggenberger M (1992) Generalized solutions to Burgers' equation. J Differential Equations97: 263–287
Biagioni H, Cadeddu L, Gramchev T (1997) Parabolic equations with conservative nonlinear term and singular initial data. In: Proc 2nd World Congress of Nonlinear Analysts, Athens, Greece, July 10–17, 1996. Nonlinear Anal TMA30: 2489–2496
Gramchev T, Cadeddu L (1996) Delta waves for the Burgers' equations. C R Acad Bulg Sci49: 15–20
Colombeau J-F (1992) Multiplication of distributions: A tool in mathematics, numerical engineering and theoretical physics. Lect Notes Math 1532. Berlin Heidelberg New York: Springer
Demengel F, Rauch J (1990) Measure valued solutions of asymptotically homogeneous semilinear hyperbolic systems in one space variable. Proc Edinb Math Soc33: 443–460
Demengel F, Serre D (1991) Nonvanishing singular parts of measure valued solutions for scalar hyperbolic equations. Comm Partial Differential Equations16: 221–254
Dix D (1996) Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation. SIAM J Math Anal27: 708–724
Gramchev T (1991) Semilinear hyperbolic systems and equations with singular initial data. Mh Math112: 99–113
Gramchev T (1995) Entropy solutions to conservation laws with singular initial data. Nonlinear Anal TMA24: 721–733
Hopf E (1950) The partial differential equationu t +uu x =μu x x. Comm Pure Appl Math3: 201–230
Kruzkhov S (1961) The Cauchy problem in the large for certain non-linear first order differential equations. Soviet Math Dokl1: 474–478
Lax PD (1957) Hyperbolic systems of conservation laws II. Comm Pure Appl Mat10: 537–566
Liu T.-P., Pierre M (1984) Source solutions and asymptotic behaviour in conservation laws. J Differential Equations51: 419–441
Olver FWJ (1974) Introduction to Asymptotics and Special Functions. New York London: Academic Press
Oberguggenberger M (1986) Weak limits of solutions to semilinear hyperbolic systems. Math Ann274: 599–607
Oberguggenberger M, Wang Y-G (1994) Generalized solutions to conservation laws. J Anal Appl13: 7–18
Oberguggenberger M, Wang Y-G (1994) Delta-waves for semilinear hyperbolic Cauchy problem. Math Nachr166: 317–327
Rauch J, Reed M (1987) Nonlinear superposition and absorption of delta waves in one space dimension. J Funct Anal73: 152–178
Author information
Authors and Affiliations
Additional information
Partially supported by 60% of MURST, Italy
Partially supported by grant MM-410/94 with MES, Bulgaria and by 40% of MURST, Italy
Rights and permissions
About this article
Cite this article
Cadeddu, L., Gramchev, T. Singular asymptotic expansions and delta waves for Burgers' equation. Monatshefte für Mathematik 126, 91–107 (1998). https://doi.org/10.1007/BF01473580
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01473580