Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Hölder space solutions of free boundary problems that arise in combustion theory

Author: G. I. Bizhanova
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 27 (2015), nomer 2.
Journal: St. Petersburg Math. J. 27 (2016), 207-235
MSC (2010): Primary 35R35
Published electronically: January 29, 2016
MathSciNet review: 3444461
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Multidimensional single-phase problems with free boundary are studies for the heat equation with derivative in the direction of the gradient of the unknown function in differential equations on the free boundary. The unique solvability of such problems is established in Hölder spaces for small times, and coercive estimates are obtained for the solutions.

References [Enhancements On Off] (What's this?)

  • 1. J. Stefan, Über einige Probleme der Theorie der Wiirmeleitung, Sitzber. Wien. Akad. Mat. Naturw. 98 (1889), no. 11a, 473-484.
  • 2. V. A. Florin, The densification of a soil medium and the filtration at a variable porosity with regard for the effect of bound water, Izv. Akad. Nauk SSSR. OTN 1 (1951), 1635-1649. (Russian)
  • 3. Augusto Visintin, Models of phase transitions, Progress in Nonlinear Differential Equations and their Applications, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1423808
  • 4. N. A. Avdonin, \cyr Matematicheskoe opisanie protsessov kristallizatsii, “Zinatne”, Riga, 1980 (Russian). MR 579349
  • 5. G. Caginalp, Conserved-phase field system: Implication for kinetic undercooling, Phis. Rev. B 38 (1988), 789-791.
  • 6. Augusto Visintin, Differential models of hysteresis, Applied Mathematical Sciences, vol. 111, Springer-Verlag, Berlin, 1994. MR 1329094
  • 7. F. A. Williams, Combustion theory, 2-nd. ed., Benjamin/Cummnings, Menlo Park, California, 1985.
  • 8. M. Muskat, The flow of homogeneous fluids through porous media, Michigan, 1937.
  • 9. N. N. Verigin, Injection of binding solutions into bedrock in order to increase the watertightness of hydrotechnical machinery, Izv. Akad. Nauk SSSR. OTN 5 (1952), 674-687. (Russian)
  • 10. I. I. Danilyuk, On the crystallization process in pattern formation, Math. Phys., vyp. 17, Naukova Dumka, Kiev, 1975, pp. 99-111. (Russian)
  • 11. L. S. Leibenzon, To the problem of globe solidification from the initial melted state, Izv. Akad. Nauk SSSR Ser. Geograf. Geofiz. 6 (1939), 625-660. (Russian)
  • 12. A. N. Tichonov, E. A. Lubimova and V. K. Vlasov, On the evolution of melting zones in the thermal history of the Earth, Dokl. Akad. Nauk SSSR. Ser. Geofiz. 188 (1969), no. 2, 338-342. (Russian)
  • 13. E. A. Lubimova, Thermics of the Earth and Moon, Nauka, Moscow, 1968. (Russian)
  • 14. S. Correra, A. Fasano, L. Fusi, and M. Primicerio, Modelling wax diffusion in crude oils: The cold finger device, Appl. Math. Modelling 31 (2007), 2286-2298.
  • 15. Avner Friedman, Conservation laws in mathematical biology, Discrete Contin. Dyn. Syst. 32 (2012), no. 9, 3081–3097. MR 2912072,
  • 16. Borys V. Bazaliy and Avner Friedman, A free boundary problem for an elliptic-parabolic system: application to a model of tumor growth, Comm. Partial Differential Equations 28 (2003), no. 3-4, 517–560. MR 1976462,
  • 17. Ei-ichi Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J. (2) 33 (1981), no. 3, 297–335. MR 633045,
  • 18. G. I. Bizhanova and V. A. Solonnikov, On problems with free boundaries for second-order parabolic equations, Algebra i Analiz 12 (2000), no. 6, 98–139 (Russian); English transl., St. Petersburg Math. J. 12 (2001), no. 6, 949–981. MR 1816513
  • 19. Arshak Petrosyan, On existence and uniqueness in a free boundary problem from combustion, Comm. Partial Differential Equations 27 (2002), no. 3-4, 763–789. MR 1900562,
  • 20. M. Pogosyan and R. Teĭmurazyan, A one-phase parabolic problem with a free boundary in a convex ring, Izv. Nats. Akad. Nauk Armenii Mat. 44 (2009), no. 3, 67–84 (Russian, with English and Russian summaries); English transl., J. Contemp. Math. Anal. 44 (2009), no. 3, 192–204. MR 2650566,
  • 21. Inwon C. Kim, A free boundary problem arising in flame propagation, J. Differential Equations 191 (2003), no. 2, 470–489. MR 1978386,
  • 22. Tung To, A free-boundary problem for the evolution 𝑝-Laplacian equation with a combustion boundary condition, Calc. Var. Partial Differential Equations 35 (2009), no. 2, 239–262. MR 2481824,
  • 23. H. Carslaw and J. Jaeger, Conduction of heat in solids, Oxford Univ. Press, New York, 1959. MR 0959730 (89f:80004)
  • 24. G. I. Bizhanova and J. F. Rodrigues, Classical solutions to parabolic systems with free boundary of Stefan type, Adv. Differential Equations 10 (2005), no. 12, 1345–1388. MR 2175009
  • 25. Galina I. Bizhanova, On the Stefan problem with a small parameter, Parabolic and Navier-Stokes equations. Part 1, Banach Center Publ., vol. 81, Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 43–63. MR 2549322,
  • 26. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, \cyr Lineĭnye i kvazilineĭnye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
  • 27. G. I. Bizhanova and V. A. Solonnikov, On the solvability of an initial-boundary value problem for a second-order parabolic equation with a time derivative in the boundary condition in a weighted Hölder space of functions, Algebra i Analiz 5 (1993), no. 1, 109–142 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 97–124. MR 1220491
  • 28. G. I. Bizhanova, On the solvability of multidimensional single-phase problems with free boundaries in Hölder spaces, Izv. Minister. Nauki Akad. Nauk Resp. Kaz. Ser. Fiz.-Mat. 5 (1996), 28–39 (Russian, with English, Russian and Kazakh summaries). MR 1641521
  • 29. G. I. Bizhanova, Investigation of the solvability in a weighted Hölder function space of the multidimensional two-phase Stefan problem and the multidimensional Florin nonstationary filtration problem for second-order parabolic equations (Cauchy-Stefan and Cauchy-Florin problems), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 213 (1994), no. Kraev. Zadachi Mat. Fiz. Smezh. Voprosy Teor. Funktsiĭ. 25, 14–47, 224 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 84 (1997), no. 1, 823–844. MR 1329308,
  • 30. G. I. Bizhanova, Solution in a weighted Hölder function space of multidimensional two-phase Stefan and Florin problems for second-order parabolic equations in a bounded domain, Algebra i Analiz 7 (1995), no. 2, 46–76 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 2, 217–241. MR 1347512

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35R35

Retrieve articles in all journals with MSC (2010): 35R35

Additional Information

G. I. Bizhanova
Affiliation: Institute of Mathematics and Mathematical Modeling, Pushkin str. 125, Almaty, Kazakhstan

Keywords: Free boundary problems, heat equation, H\"older spaces, unique solvability, coercive estimates for solutions
Received by editor(s): October 10, 2014
Published electronically: January 29, 2016
Dedicated: Dedicated to Professor Vsevolod Alekseevich Solonnikov on the occasion of his anniversary
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society