Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MAXIMAL DOMAINS OF SOLUTIONS FOR ANALYTIC QUASILINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER

  • Han, Chong-Kyu (Research Institute of Mathematics Seoul National University) ;
  • Kim, Taejung (Department of Mathematical Education Korea National University of Education)
  • Received : 2022.01.25
  • Accepted : 2022.09.06
  • Published : 2022.11.01
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Abstract

We study the real-analytic continuation of local real-analytic solutions to the Cauchy problems of quasi-linear partial differential equations of first order for a scalar function. By making use of the first integrals of the characteristic vector field and the implicit function theorem we determine the maximal domain of the analytic extension of a local solution as a single-valued function. We present some examples including the scalar conservation laws that admit global first integrals so that our method is applicable.

Keywords

Acknowledgement

The authors were supported by NRF-Republic of Korea with grants 0450-20210049 (C.-K. Han) and 2018R1D1A3B07043346 (T. Kim), respectively.

References

  1. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGrawHill Book Co., Inc., New York, 1955.
  2. R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, NY, 1953.
  3. L. C. Evans, Partial differential equations, second edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. https://doi.org/10.1090/gsm/019
  4. C.-K. Han and H. Kim, Invariant submanifolds for affine control systems, Ann. Polon. Math. 124 (2020), no. 1, 61-73. https://doi.org/10.4064/ap190327-16-10
  5. C.-K Han and J. Park, Method of characteristics and first integrals for systems of quasilinear partial differential equations of first order, Sci. China Math. 58 (2015), no. 8, 1665-1676. https://doi.org/10.1007/s11425-014-4942-8
  6. S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, second edition, Birkhauser Advanced Texts: Basler Lehrbucher., Birkhauser Boston, Inc., Boston, MA, 2002. https://doi.org/10.1007/978-0-8176-8134-0
  7. P. D. Lax, The formation and decay of shock waves, Amer. Math. Monthly 79 (1972), 227-241. https://doi.org/10.2307/2316618
  8. F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman & Co., Glenview, IL, 1971.