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 \[ {\partial _t}u + \nabla \cdot f\left ( u \right ) = {- \frac {1}{\epsilon }g\left ( u \right ) + \epsilon \Delta u}, \qquad {x \in \mathbb {R}{^n}}, {u \in \mathbb {R}} \qquad \left ( 1 \right )\] 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 \[ \epsilon \left ( {k + \frac {{\nabla \mu }}{\mu } \cdot T} \right )\], 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.
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)
- G. Barles, L. C. Evans, and P. E. Souganidis, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J. 61 (1990), no. 3, 835–858. MR 1084462, DOI https://doi.org/10.1215/S0012-7094-90-06132-0
- Lia Bronsard and Robert V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990), no. 8, 983–997. MR 1075075, DOI https://doi.org/10.1002/cpa.3160430804
- Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 485012
- Gunduz Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal. 92 (1986), no. 3, 205–245. MR 816623, DOI https://doi.org/10.1007/BF00254827
- Xu-Yan Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J. 21 (1991), no. 1, 47–83. MR 1091432
- E. De Giorgi, New conjectures on flow by mean curvature, Nonlinear variational problems and partial differential equations (Isola d’Elba, 1990) Pitman Res. Notes Math. Ser., vol. 320, Longman Sci. Tech., Harlow, 1995, pp. 120–128. MR 1330007
- Lawrence C. Evans, Regularity for fully nonlinear elliptic equations and motion by mean curvature, Viscosity solutions and applications (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997, pp. 98–133. MR 1462701, DOI https://doi.org/10.1007/BFb0094296
- L. C. Evans, H. M. Soner, and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), no. 9, 1097–1123. MR 1177477, DOI https://doi.org/10.1002/cpa.3160450903
- Hai Tao Fan and Jack K. Hale, Large time behavior in inhomogeneous conservation laws, Arch. Rational Mech. Anal. 125 (1993), no. 3, 201–216. MR 1245070, DOI https://doi.org/10.1007/BF00383219
- Hai Tao Fan and Jack K. Hale, Attractors in inhomogeneous conservation laws and parabolic regularizations, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1239–1254. MR 1270661, DOI https://doi.org/10.1090/S0002-9947-1995-1270661-9
- Paul C. Fife and Ling Hsiao, The generation and propagation of internal layers, Nonlinear Anal. 12 (1988), no. 1, 19–41. MR 924750, DOI https://doi.org/10.1016/0362-546X%2888%2990010-7
- Paul C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 981594
- Haitao Fan, Shi Jin, and Zhen-huan Teng, Zero reaction limit for hyperbolic conservation laws with source terms, J. Differential Equations 168 (2000), no. 2, 270–294. Special issue in celebration of Jack K. Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998). MR 1808451, DOI https://doi.org/10.1006/jdeq.2000.3887
H. Fan and S. Jin, Wave patterns and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms, preprint (2000)
- Jörg Härterich, Heteroclinic orbits between rotating waves in hyperbolic balance laws, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 3, 519–538. MR 1693629, DOI https://doi.org/10.1017/S0308210500021491
- Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), no. 2, 417–461. MR 1237490
- Athanasios N. Lyberopoulos, A Poincaré-Bendixson theorem for scalar balance laws, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 3, 589–607. MR 1286920, DOI https://doi.org/10.1017/S0308210500028791
- C. Mascia, Travelling wave solutions for a balance law, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 3, 567–593. MR 1453282, DOI https://doi.org/10.1017/S0308210500029917
- Jacob Rubinstein, Peter Sternberg, and Joseph B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49 (1989), no. 1, 116–133. MR 978829, DOI https://doi.org/10.1137/0149007
- Carlo Sinestrari, Asymptotic profile of solutions of conservation laws with source, Differential Integral Equations 9 (1996), no. 3, 499–525. MR 1371704
- Halil Mete Soner, Ginzburg-Landau equation and motion by mean curvature. II. Development of the initial interface, J. Geom. Anal. 7 (1997), no. 3, 477–491. MR 1674800, DOI https://doi.org/10.1007/BF02921629
- Panagiotis E. Souganidis, Front propagation: theory and applications, Viscosity solutions and applications (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997, pp. 186–242. MR 1462703, DOI https://doi.org/10.1007/BFb0094298
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)
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)
L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one dimension, Comm. Pure Appl. Math. 43, 983–997 (1990)
K. A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal. 92, no. 3, 205–245 (1986)
X.-Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J. 21, 47–83 (1991)
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
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
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)
H. Fan and J. K. Hale, Large-time behavior in inhomogeneous conservation laws, Arch. Rational Mech. Anal. 125, 201–216 (1993)
H. Fan and J. K. Hale, Attractors in inhomogeneous conservation laws and parabolic regularizations, Trans. Amer. Math. Soc. 347, 1239–1254 (1995)
P. C. Fife and L. Hsiao, The generation and propagation of internal layers, Nonlinear Anal. 12, no. 1, 19–41 (1988)
P. C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 53, SIAM, Philadelphia (1988)
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)
H. Fan and S. Jin, Wave patterns and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms, preprint (2000)
J. Härterich, Heteroclinic orbits between rotating waves in hyperbolic balance laws, Proc. Roy. Soc. Edinburgh Sect. A 129, no. 3, 519–538 (1999)
T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38, no. 2, 417–461 (1993)
A. N. Lyberopoulos, A Poincarè-Bendixson theorem for scalar conservation laws, Proc. Roy Soc. Edinburgh, 124A, 589–607 (1994)
C. Mascia, Traveling wave solutions for a balance law, Proc. Roy. Soc. Edinburgh 127A, 567–593 (1997)
J. Rubinstein, P. Sternberg, and J. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49, 116–133 (1989)
C. Sinestrari, Asymptotic profile of solutions of conservation laws with source, Differential Integral Equations 9, 499–525 (1996)
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)
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
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35K57,
35B25,
35L65
Retrieve articles in all journals
with MSC:
35K57,
35B25,
35L65
Additional Information
Article copyright:
© Copyright 2003
American Mathematical Society