## Stability of fronts for a regularization of the Burgers equation.(English)Zbl 1160.37029

In the previous work [J. Nonl. Sci. 16, No. 6, 615–638 (2006; Zbl 1108.35107)] the following regularization of the Leray-type for the Burgers equation was introduced $v_t+uv_x=0, v=u-\alpha^2 u_{xx},\eqno{(1)}$ where $$\alpha>0$$ is a constant that has the dimension of the length. It was shown that the system (1) is globally well-posed with initial data $$v(x,0)$$ in the Sobolev space $$W^{2,1}(\mathbb{R})$$, and also the strong convergence of the solutions $$u^{\alpha}(x,t)$$ at $$\alpha\rightarrow 0$$ to the entropy solution of the inviscid Burgers equation. In this article the authors prove stability (instability) in some sense for monotone decreasing (monotone increasing) fronts, i.e. travelling waves which are monotonic profiles connecting a left state to a right state. These two types of travelling fronts correspond, respectively, to viscous shocks and rarefaction waves.

### MSC:

 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems

Zbl 1108.35107
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### References:

 [1] H. S. Bhat and R. C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci. 16 (2006), no. 6, 615 – 638. · Zbl 1108.35107 [2] Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661 – 1664. · Zbl 0972.35521 [3] Adrian Constantin and Walter A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000), no. 5, 603 – 610. , https://doi.org/10.1002/(SICI)1097-0312(200005)53:53.3.CO;2-C · Zbl 1049.35149 [4] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci. 12 (2002), no. 4, 415 – 422. · Zbl 1022.35053 [5] H. R. Dullin, G. A. Gottwald, and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett. 87 (2001), no. 19, 194501-1-4. [6] Holger R. Dullin, Georg A. Gottwald, and Darryl D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res. 33 (2003), no. 1-2, 73 – 95. In memoriam Prof. Philip Gerald Drazin 1934 – 2002. · Zbl 1032.76518 [7] H. R. Dullin, G. A. Gottwald, and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D 190 (2004), no. 1-2, 1 – 14. · Zbl 1050.76008 [8] A. Degasperis, D. D. Holm, and A. N. W. Hone, Integrable and non-integrable equations with peakons, Nonlinear physics: theory and experiment, II (Gallipoli, 2002) World Sci. Publ., River Edge, NJ, 2003, pp. 37 – 43. · Zbl 1053.37039 [9] A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and perturbation theory (Rome, 1998) World Sci. Publ., River Edge, NJ, 1999, pp. 23 – 37. · Zbl 0963.35167 [10] Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325 – 344. · Zbl 0631.35058 [11] Darryl D. Holm and Andrew N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonlinear Math. Phys. 12 (2005), no. suppl. 1, 380 – 394. With an appendix by H. W. Braden and J. G. Byatt-Smith. · Zbl 1362.35316 [12] Darryl D. Holm and Martin F. Staley, Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Phys. Lett. A 308 (2003), no. 5-6, 437 – 444. · Zbl 1010.35066 [13] Darryl D. Holm and Martin F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst. 2 (2003), no. 3, 323 – 380. · Zbl 1088.76531 [14] Andrew N. W. Hone and Jing Ping Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems 19 (2003), no. 1, 129 – 145. · Zbl 1020.35096 [15] Arieh Iserles, A first course in the numerical analysis of differential equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1996. · Zbl 1171.65060 [16] P. Kurasov, Distribution theory for discontinuous test functions and differential operators with generalized coefficients, J. Math. Anal. Appl. 201 (1996), no. 1, 297 – 323. · Zbl 0878.46030 [17] Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193 – 248 (French). · JFM 60.0726.05 [18] K. Mohseni, H. Zhao, and J. E. Marsden, Shock regularization for the Burgers equation, 44th AIAA Aerospace Sciences Meeting and Exhibit (Reno, NV), January 2006, AIAA Paper 2006-1516. [19] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312 – 355. · Zbl 0344.35051
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