# American Institute of Mathematical Sciences

December  2012, 32(12): 4069-4110. doi: 10.3934/dcds.2012.32.4069

## Dafermos regularization of a diffusive-dispersive equation with cubic flux

 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205 2 Department of Mathematics, Shepherd University, Shepherdstown, WV 25443-5000, United States

Received  June 2011 Revised  June 2012 Published  August 2012

We study existence and spectral stability of stationary solutions of the Dafermos regularization of a much-studied diffusive-dispersive equation with cubic flux. Our study includes stationary solutions that corresponds to Riemann solutions consisting of an undercompressive shock wave followed by a compressive shock wave. We use geometric singular perturbation theory (1) to construct the solutions, and (2) to show that asmptotically, there are no large eigenvalues, and any order-one eigenvalues must be near $-1$ or a certain number $\lambda^*$. We give numerical evidence that $\lambda^*$ is also $-1$. Finally, we use pseudoexponential dichotomies to show that in a space of exponentially decreasing functions, the essential spectrum is contained in Re$\lambda \le -\delta <0$.
Citation: Stephen Schecter, Monique Richardson Taylor. Dafermos regularization of a diffusive-dispersive equation with cubic flux. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4069-4110. doi: 10.3934/dcds.2012.32.4069
##### References:
 [1] A. Azevedo, D. Marchesin, B. J. Plohr and K. Zumbrun, Nonuniqueness of solutions of Riemann problems, Zeit. angew. Math. Phys., 47 (1996), 977-998. doi: 10.1007/BF00920046. [2] C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation lawsby the viscosity method, Arch. Ration. Mech. Anal., 52 (1973), 1-9. doi: 10.1007/BF00249087. [3] J. Dodd, Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation,, Electron. J. Differential Equations, 2007 (). [4] P. Howard and K. Zumbrun, Pointwise estimates and stability for dispersive-diffusive shock waves, Arch. Ration. Mech. Anal., 155 (2000), 85-169. doi: 10.1007/s002050000110. [5] P. Howard and K. Zumbrun, The Evans function and stability criteria for degenerate viscous shock waves, Discrete Contin. Dyn. Syst., 10 (2004), 837-855. doi: 10.3934/dcds.2004.10.837. [6] D. Jacobs, B. McKinney and M. Shearer, Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation, J. Differential Equations, 116 (1995), 448-467. doi: 10.1006/jdeq.1995.1043. [7] T. J. Kaper and C. K. R. T. Jones, A primer on the exchange lemma for fast-slow systems. Multiple-time-scale dynamical systems (Minneapolis, MN, 1997), 65-87, IMA Vol. Math. Appl., 122, Springer, New York, 2001. [8] C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, 1994), 44-118, Lecture Notes in Math. 1609, Springer, Berlin, 1995. [9] C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds withdifferential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), 64-89. doi: 10.1006/jdeq.1994.1025. [10] C. K. R. T. Jones and S.-K. Tin, Generalized exchange lemmas and orbits heteroclinic to invariant manifolds, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 967-1023. doi: 10.3934/dcdss.2009.2.967. [11] K. T. Joseph and P. G. LeFloch, Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions, Analytical Approaches to Multidimensional Balance Laws, 143-172, Nova Sci. Publ., New York, 2006. [12] K. T. Joseph and P. G. LeFloch, Singular limits in phase dynamics with physical viscosity and capillarity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1287-1312. doi: 10.1017/S030821050600093X. [13] P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves," Lectures in Mathematics ETH Zuich, Birkhauser, Basel, 2002. [14] P. G. LeFloch and C. Rohde, Zero diffusion-dispersion limits for self-similar Riemann solutions to hyperbolic systems of conservation laws, Indiana Univ. Math. J., 50 (2001), 1707-743. doi: 10.1512/iumj.2001.50.2057. [15] X.-B. Lin, Analytic semigroup generated by the linearization of a Riemann-Dafermos solution, Dyn. Partial Differ. Equ., 1 (2004), 193-207. [16] X.-B. Lin, Slow eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws: an analytic approach, J. Dynam. Differential Equations, 18 (2006), 1-52. doi: 10.1007/s10884-005-9001-2. [17] X.-B. Lin and S. Schecter, Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws, SIAM J. Math. Anal., 35 (2003), 884-921. doi: 10.1137/S0036141002405029. [18] T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108. [19] W. Liu, Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws, Discrete Contin. Dyn. Syst., 10 (2004), 871-884. doi: 10.3934/dcds.2004.10.871. [20] B. Sandstede, Stability of traveling waves, in "Handbook of Dynamical Systems," Vol. 2, 983-1055, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80039-X. [21] S. Schecter, Undercompressive shock waves and the Dafermos regularization, Nonlinearity, 15 (2002), 1361-1377. doi: 10.1088/0951-7715/15/4/318. [22] S. Schecter, Eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws via geometric singular perturbation theory, J. Dynam. Differential Equations, 18 (2006), 53-101. doi: 10.1007/s10884-005-9000-3. [23] S. Schecter and P. Szmolyan, Composite waves in the Dafermos regularization, J. Dynam. Differential Equations, 16 (2004), 847-867. doi: 10.1007/s10884-004-6698-2. [24] S. Schecter and P. Szmolyan, Persistence of rarefactions under Dafermos regularization: blow-up and an exchange lemma for gain-of-stability turning points, SIAM J. Appl. Dyn. Syst., 8 (2009), 822-853. doi: 10.1137/080715305. [25] A. Szepessy and K. Zumbrun, Stability of rarefaction waves in viscous media, Arch. Ration. Mech. Anal., 133 (1996), 249-298. doi: 10.1007/BF00380894. [26] V. A. Tupčiev, On the splitting of an arbitrary discontinuity for a system of two first-order quasi-linear equations, Ž. Vyčisl. Mat. i Mat. Fiz., 4, 817-825. English translation: USSR Comput. Math. Math. Phys. 4 (1964), 36-48. [27] V. A. Tupčiev, The method of introducing a viscosity in the study of a problem of decay of a discontinuity, Dokl. Akad. Nauk SSSR, 211, 55-58. English translation: Soviet Math. Dokl. 14 (1973), 978-982. [28] A. E. Tzavaras, Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws, Arch. Ration. Mech. Anal., 135 (1996), 1-60. doi: 10.1007/BF02198434. [29] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871. doi: 10.1512/iumj.1998.47.1604.

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##### References:
 [1] A. Azevedo, D. Marchesin, B. J. Plohr and K. Zumbrun, Nonuniqueness of solutions of Riemann problems, Zeit. angew. Math. Phys., 47 (1996), 977-998. doi: 10.1007/BF00920046. [2] C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation lawsby the viscosity method, Arch. Ration. Mech. Anal., 52 (1973), 1-9. doi: 10.1007/BF00249087. [3] J. Dodd, Spectral stability of undercompressive shock profile solutions of a modified KdV-Burgers equation,, Electron. J. Differential Equations, 2007 (). [4] P. Howard and K. Zumbrun, Pointwise estimates and stability for dispersive-diffusive shock waves, Arch. Ration. Mech. Anal., 155 (2000), 85-169. doi: 10.1007/s002050000110. [5] P. Howard and K. Zumbrun, The Evans function and stability criteria for degenerate viscous shock waves, Discrete Contin. Dyn. Syst., 10 (2004), 837-855. doi: 10.3934/dcds.2004.10.837. [6] D. Jacobs, B. McKinney and M. Shearer, Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation, J. Differential Equations, 116 (1995), 448-467. doi: 10.1006/jdeq.1995.1043. [7] T. J. Kaper and C. K. R. T. Jones, A primer on the exchange lemma for fast-slow systems. Multiple-time-scale dynamical systems (Minneapolis, MN, 1997), 65-87, IMA Vol. Math. Appl., 122, Springer, New York, 2001. [8] C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, 1994), 44-118, Lecture Notes in Math. 1609, Springer, Berlin, 1995. [9] C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds withdifferential forms in singularly perturbed systems, J. Differential Equations, 108 (1994), 64-89. doi: 10.1006/jdeq.1994.1025. [10] C. K. R. T. Jones and S.-K. Tin, Generalized exchange lemmas and orbits heteroclinic to invariant manifolds, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 967-1023. doi: 10.3934/dcdss.2009.2.967. [11] K. T. Joseph and P. G. LeFloch, Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions, Analytical Approaches to Multidimensional Balance Laws, 143-172, Nova Sci. Publ., New York, 2006. [12] K. T. Joseph and P. G. LeFloch, Singular limits in phase dynamics with physical viscosity and capillarity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1287-1312. doi: 10.1017/S030821050600093X. [13] P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves," Lectures in Mathematics ETH Zuich, Birkhauser, Basel, 2002. [14] P. G. LeFloch and C. Rohde, Zero diffusion-dispersion limits for self-similar Riemann solutions to hyperbolic systems of conservation laws, Indiana Univ. Math. J., 50 (2001), 1707-743. doi: 10.1512/iumj.2001.50.2057. [15] X.-B. Lin, Analytic semigroup generated by the linearization of a Riemann-Dafermos solution, Dyn. Partial Differ. Equ., 1 (2004), 193-207. [16] X.-B. Lin, Slow eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws: an analytic approach, J. Dynam. Differential Equations, 18 (2006), 1-52. doi: 10.1007/s10884-005-9001-2. [17] X.-B. Lin and S. Schecter, Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws, SIAM J. Math. Anal., 35 (2003), 884-921. doi: 10.1137/S0036141002405029. [18] T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56 (1985), 1-108. [19] W. Liu, Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws, Discrete Contin. Dyn. Syst., 10 (2004), 871-884. doi: 10.3934/dcds.2004.10.871. [20] B. Sandstede, Stability of traveling waves, in "Handbook of Dynamical Systems," Vol. 2, 983-1055, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80039-X. [21] S. Schecter, Undercompressive shock waves and the Dafermos regularization, Nonlinearity, 15 (2002), 1361-1377. doi: 10.1088/0951-7715/15/4/318. [22] S. Schecter, Eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws via geometric singular perturbation theory, J. Dynam. Differential Equations, 18 (2006), 53-101. doi: 10.1007/s10884-005-9000-3. [23] S. Schecter and P. Szmolyan, Composite waves in the Dafermos regularization, J. Dynam. Differential Equations, 16 (2004), 847-867. doi: 10.1007/s10884-004-6698-2. [24] S. Schecter and P. Szmolyan, Persistence of rarefactions under Dafermos regularization: blow-up and an exchange lemma for gain-of-stability turning points, SIAM J. Appl. Dyn. Syst., 8 (2009), 822-853. doi: 10.1137/080715305. [25] A. Szepessy and K. Zumbrun, Stability of rarefaction waves in viscous media, Arch. Ration. Mech. Anal., 133 (1996), 249-298. doi: 10.1007/BF00380894. [26] V. A. Tupčiev, On the splitting of an arbitrary discontinuity for a system of two first-order quasi-linear equations, Ž. Vyčisl. Mat. i Mat. Fiz., 4, 817-825. English translation: USSR Comput. Math. Math. Phys. 4 (1964), 36-48. [27] V. A. Tupčiev, The method of introducing a viscosity in the study of a problem of decay of a discontinuity, Dokl. Akad. Nauk SSSR, 211, 55-58. English translation: Soviet Math. Dokl. 14 (1973), 978-982. [28] A. E. Tzavaras, Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws, Arch. Ration. Mech. Anal., 135 (1996), 1-60. doi: 10.1007/BF02198434. [29] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871. doi: 10.1512/iumj.1998.47.1604.
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