Front motion in multi-dimensional viscous conservation laws with stiff source terms driven by mean curvature and variation of front thickness

Authors:
Haitao Fan and Shi Jin

Journal:
Quart. Appl. Math. **61** (2003), 701-721

MSC:
Primary 35K57; Secondary 35B25, 35L65

DOI:
https://doi.org/10.1090/qam/2019619

MathSciNet review:
MR2019619

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The bistable reaction-diffusion-convection equation

**n**of the front, and

**T**is a vector tangential to the front. Both and

**T**are elliptic operators, contributing to the shrinkage of closed curves. An ellipse in is found to preserve its shape while shrinking.

**[AC]**S. M. Allen and J. M. Cahn,*A macroscopic theory for antiphase boundary motion and its applications to antiphase domain coarsening*, Acta Metal.**27**, 1085-1095 (1979)**[BES]**G. Barles, L. C. Evans, and P. E. Souganidis,*Wavefront propagation for reaction-diffusion systems of PDE*, Duke Math. J.**61**, no. 3, 835-858 (1990) MR**1084462****[BK]**L. Bronsard and R. V. Kohn,*On the slowness of phase boundary motion in one dimension*, Comm. Pure Appl. Math.**43**, 983-997 (1990) MR**1075075****[Br]**K. A. Brakke,*The motion of a surface by its mean curvature*, Mathematical Notes, vol.**20**, Princeton University Press, Princeton, N.J., 1978 MR**485012****[Ca]**G. Caginalp,*An analysis of a phase field model of a free boundary*, Arch. Rational Mech. Anal.**92**, no. 3, 205-245 (1986) MR**816623****[Ch]**X.-Y. Chen,*Dynamics of interfaces in reaction diffusion systems*, Hiroshima Math. J.**21**, 47-83 (1991) MR**1091432****[D]**E. De Giorgi,*New conjectures on flow by mean curvature*, Nonlinear variational problems and partial differential equations (Isola d'Elba, 1990), 120-128, Pitman Res. Notes Math. Ser., vol. 320, Longman Sci. Tech., Harlow, 1995 MR**1330007****[E]**L. C. Evans,*Regularity for fully nonlinear elliptic equations and motion by mean curvature*, Viscosity solutions and applications (Montecantini Terme, 1995), 98-133, Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997 MR**1462701****[ESS]**L. C. Evans, H. M. Soner, and P. E. Souganidis,*Phase transitions and generalized motion by mean curvature*, Comm. Pure Appl. Math.**45**, no. 9, 1097-1123 (1992) MR**1177477****[FH1]**H. Fan and J. K. Hale,*Large-time behavior in inhomogeneous conservation laws*, Arch. Rational Mech. Anal.**125**, 201-216 (1993) MR**1245070****[FH2]**H. Fan and J. K. Hale,*Attractors in inhomogeneous conservation laws and parabolic regularizations*, Trans. Amer. Math. Soc.**347**, 1239-1254 (1995) MR**1270661****[FHs]**P. C. Fife and L. Hsiao,*The generation and propagation of internal layers*, Nonlinear Anal.**12**, no. 1, 19-41 (1988) MR**924750****[Fife]**P. C. Fife,*Dynamics of internal layers and diffusive interfaces*, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 53, SIAM, Philadelphia (1988) MR**981594****[FJT]**H. Fan, S. Jin, and Z.-H. Teng,*Zero reaction limit for hyperbolic conservation laws with source terms*, J. Diff. Eqs.**168**, 270-294 (2000) MR**1808451****[FJ]**H. Fan and S. Jin,*Wave patterns and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms*, preprint (2000)**[Har]**J. Härterich,*Heteroclinic orbits between rotating waves in hyperbolic balance laws*, Proc. Roy. Soc. Edinburgh Sect. A**129**, no. 3, 519-538 (1999) MR**1693629****[Il]**T. Ilmanen,*Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature*, J. Differential Geom.**38**, no. 2, 417-461 (1993) MR**1237490****[Lyb]**A. N. Lyberopoulos,*A Poincarè-Bendixson theorem for scalar conservation laws*, Proc. Roy Soc. Edinburgh,**124**A, 589-607 (1994) MR**1286920****[Mas]**C. Mascia,*Traveling wave solutions for a balance law*, Proc. Roy. Soc. Edinburgh**127**A, 567-593 (1997) MR**1453282****[RSK]**J. Rubinstein, P. Sternberg, and J. Keller,*Fast reaction, slow diffusion, and curve shortening*, SIAM J. Appl. Math.**49**, 116-133 (1989) MR**978829****[Sin2]**C. Sinestrari,*Asymptotic profile of solutions of conservation laws with source*, Differential Integral Equations**9**, 499-525 (1996) MR**1371704****[Son]**H. M. Soner,*Ginzburg-Landau equation and motion by mean curvature. II. Development of the initial interface*, J. Geom. Anal.**7**, no. 3, 477-491 (1997) MR**1674800****[Sou]**P. E. Souganidis,*Front propagation: theory and applications*, Viscosity solutions and applications (Montecatini Terme, 1995), 186-242, Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997 MR**1462703**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35K57,
35B25,
35L65

Retrieve articles in all journals with MSC: 35K57, 35B25, 35L65

Additional Information

DOI:
https://doi.org/10.1090/qam/2019619

Article copyright:
© Copyright 2003
American Mathematical Society