Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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Abstract | References | Similar Articles | Additional Information

Abstract: The bistable reaction-diffusion-convection equation

$\displaystyle {\partial _t}u + \nabla \cdot f\left( u \right) = {- \frac{1}{\ep... ...a u}, \qquad {x \in \mathbb{R}{^n}}, {u \in \mathbb{R}} \qquad \left( 1 \right)$

is considered. Stationary traveling waves of the above equation are proved to exist when $ f\left( u \right)$ is symmetric and $ g\left( u \right)$ is antisymmetric about $ u = 0$. Solutions of initial value problems tend to almost piecewise constant functions within $ O\left( 1 \right)\epsilon $ time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts is studied by asymptotic expansion. The equation for the motion of the front is obtained. In the case of $ f = b{u^2}$ and $ g\left( u \right) = au\left( 1 - {u^2} \right)$, where $ b \in {\mathbb{R}^{n}}$ and $ 0 < a \in \mathbb{R}$ are constants, the front motion equation takes a more explicit form, showing that the front's speed is

$\displaystyle \epsilon \left( {k + \frac{{\nabla \mu }}{\mu } \cdot T} \right)$

, where $ \kappa $ is the mean curvature of the front, $ \mu $ is the width of the planar traveling of (1) in the normal direction n of the front, and T is a vector tangential to the front. Both $ \kappa $ and $ \nabla \mu /\mu \cdot T$ T are elliptic operators, contributing to the shrinkage of closed curves. An ellipse in $ {\mathbb{R}^{2}}$ is found to preserve its shape while shrinking.

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

  • [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, 124A, 589-607 (1994) MR 1286920
  • [Mas] C. Mascia, Traveling wave solutions for a balance law, Proc. Roy. Soc. Edinburgh 127A, 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

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DOI: https://doi.org/10.1090/qam/2019619
Article copyright: © Copyright 2003 American Mathematical Society

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