×

Hamiltonian regularisation of the unidimensional barotropic Euler equations. (English) Zbl 07446161

Summary: Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh (2018). This system is Galilean invariant, linearly non-dispersive and conserves formally an \(H^1\)-like energy. In this paper, we extend this regularisation in two directions. First, we consider the more general barotropic Euler system, the shallow water equations being formally a very special case. Second, we introduce a class regularisations, showing thus that this Hamiltonian regularisation of Clamond and Dutykh (2018) is not unique. Considering the high-frequency approximation of this regularisation, we obtain a new two-component Hunter-Saxton system. We prove that both systems – the regularised barotropic Euler system and the two-component Hunter-Saxton system – are locally (in time) well-posed, and, if singularities appear in finite time, they are necessary in the first derivatives.

MSC:

35Qxx Partial differential equations of mathematical physics and other areas of application
35Lxx Hyperbolic equations and hyperbolic systems
35Bxx Qualitative properties of solutions to partial differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Wagner, W.; Pruß, A., The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, J. Phys. Chem. Ref. Data, 31, 2, 387-535 (1995)
[2] Bianchini, S.; Bressan, A., Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math., 223-342 (2005) · Zbl 1082.35095
[3] Crandall, M. G.; Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277, 1, 1-42 (1983) · Zbl 0599.35024
[4] Hayes, B. T.; LeFloch, P. G., Nonclassical shocks and kinetic relations: strictly hyperbolic systems, SIAM J. Math. Anal., 31, 5, 941-991 (2000) · Zbl 0953.35095
[5] Kondo, C. I.; LeFloch, P. G., Zero diffusion-dispersion limits for scalar conservation laws, SIAM J. Math. Anal., 33, 6, 1320-1329 (2002) · Zbl 1055.35072
[6] Lax, P. D.; Levermore, C. D., The small dispersion limit of the KdV equations I, Comm. Pure Appl. Math., 3, 253-290 (1983) · Zbl 0532.35067
[7] Lax, P. D.; Levermore, C. D., The small dispersion limit of the KdV equations II, Comm. Pure Appl. Math., 5, 571-593 (1983) · Zbl 0527.35073
[8] Lax, P. D.; Levermore, C. D., The small dispersion limit of the KdV equations III, Comm. Pure Appl. Math., 6, 809-829 (1983) · Zbl 0527.35074
[9] VonNeumann, J.; Richtmyer, R. D., A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21, 3, 232-237 (1950) · Zbl 0037.12002
[10] Bhat, H. S.; Fetecau, R. C., A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci., 16, 6, 615-638 (2006) · Zbl 1108.35107
[11] Bhat, H. S.; Fetecau, R. C., The Riemann problem for the Leray-Burgers equation, J. Differential Equations, 246, 3957-3979 (2009) · Zbl 1177.35136
[12] Bhat, H. S.; Fetecau, R. C., On a regularization of the compressible Euler equations for an isothermal gas, J. Math. Anal. Appl., 358, 1, 168-181 (2009) · Zbl 1178.35037
[13] Bhat, H. S.; Fetecau, R. C.; Goodman, J., A Leray-type regularization for the isentropic Euler equations, Nonlinearity, 20, 9, 2035-2046 (2007) · Zbl 1139.35072
[14] Camassa, R.; Chiu, P.-H.; Lee, L.; Sheu, T. W.H., Viscous and inviscid regularizations in a class of evolutionary partial differential equations, J. Comput. Phys., 229, 6676-6687 (2010) · Zbl 1426.35184
[15] Norgard, G. J.; Mohseni, K., An examination of the homentropic Euler equations with averaged characteristics, J. Differential Equations, 248, 574-593 (2010) · Zbl 1183.76791
[16] Norgard, G. J.; Mohseni, K., A new potential regularization of the one-dimensional Euler and homentropic Euler equations, Multiscale Model. Simul., 8, 4, 1212-1243 (2010) · Zbl 1383.76394
[17] Clamond, D.; Dutykh, D., Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler) equations, Commun. Nonlinear Sci. Numer. Simul., 55, 237-247 (2018) · Zbl 1460.76076
[18] Pu, Y.; Pego, R. L.; Dutykh, D.; Clamond, D., Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations, Commun. Math. Sci., 16, 5, 1361-1378 (2018) · Zbl 1406.35193
[19] Liu, J.-L.; Pego, R. L.; Pu, Y., Well-posedness and derivative blow-up for a dispersionless regularized shallow water system, Nonlinearity, 32, 11, 4346-4376 (2019) · Zbl 1428.35375
[20] Guelmame, B.; Junca, S.; Clamond, D.; Pego, R., Global weak solutions of a Hamiltonian regularised Burgers equation (2020), hal-02478872
[21] Guelmame, B., On a Hamiltonian regularization of scalar conservation laws (2020), hal-02512810
[22] Clamond, D.; Dutykh, D.; Mitsotakis, D., Hamiltonian regularisation of shallow water equations with uneven bottom, J. Phys. A, 52, 42, Article 42LT01 pp. (2019)
[23] Clamond, D.; Dutykh, D.; Mitsotakis, D., Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics, Commun. Nonlinear Sci. Numer. Simul., 45, 245-257 (2017) · Zbl 1459.76020
[24] Alinhac, S., Blowup for nonlinear hyperbolic equations (2013), Springer Science & Business Media
[25] Alinhac, S.; Gérard, P., Pseudo-differential operators and the Nash-Moser theorem, Amer. Math. Soc., 82 (2007) · Zbl 1121.47033
[26] Israwi, S., Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74, 1, 81-93 (2011) · Zbl 1381.86012
[27] Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (2012), Springer Science & Business Media
[28] Serrin, J., Mathematical principles of classical fluid mechanics, (Flugge, S.; Truedell, C., Fluid Dynamics I, Volume VIII/1 of Encyclopaedia of Physics (1959), Springer: Springer Verlag), 125-263
[29] Dolzhansky, F. V., Fundamentals of Geophysical Hydrodynamics (2003), Springer
[30] Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics (2016), Springer · Zbl 1364.35003
[31] Lax, P. D., Shock waves and entropy, (Contributions to Nonlinear Functional Analysis (1971)), 603-634
[32] Clamond, D.; Dutykh, D., Fast accurate computation of the fully nonlinear solitary surface gravity waves, Comp. Fluids, 84, 35-38 (2013) · Zbl 1290.76018
[33] Li, Y. A., Linear stability of solitary waves of the Green-Naghdi equations, Comm. Pure Appl. Math., 54, 5, 501-536 (2001) · Zbl 1032.35158
[34] Nutku, Y., On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure, J. Math. Phys., 28, 11, 2579-2585 (1987) · Zbl 0662.35084
[35] Pu, Y., Weakly Singular Waves and Blow-Up for a Regularization of the Shallow-Water System (2018), Carnegie Mellon University, (Ph.D. thesis)
[36] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 7, 891-907 (1988) · Zbl 0671.35066
[37] Constantin, A.; Molinet, L., The initial value problem for a generalized Boussinesq equation, Differ. Integral Equ. Appl., 15, 9, 1061-1072 (2002) · Zbl 1161.35445
[38] Hunter, J. K.; Saxton, R., Dynamics of director fields, SIAM J. Appl. Math., 51, 6, 1498-1521 (1991) · Zbl 0761.35063
[39] Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations, (Spectral Theory and Differential Equations (1975), Springer), 25-70
[40] Liu, J.; Yin, Z., On the Cauchy problem of a periodic 2-component \(\mu \)-Hunter-Saxton system, Nonlinear Anal. TMA, 75, 1, 131-142 (2012) · Zbl 1228.35093
[41] Liu, J.; Yin, Z., On the Cauchy problem of a weakly dissipative \(\mu \)-Hunter-Saxton equation, Annales de L’Institut Henri Poincare (C) Non Linear Analysis, 31, 267-279 (2014) · Zbl 1302.35320
[42] Moon, B.; Liu, Y., Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system, J. Differential Equations, 253, 1, 319-355 (2012) · Zbl 1248.35052
[43] Yin, Z., On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36, 1, 272-283 (2004) · Zbl 1151.35321
[44] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 2, 215-239 (2007) · Zbl 1105.76013
[45] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5, 1, 1-27 (2007) · Zbl 1139.35378
[46] Grunert, K.; Holden, H.; Raynaud, X., Global solutions for the two-component Camassa-Holm system, Comm. Partial Differential Equations, 37, 12, 2245-2271 (2012) · Zbl 1258.35176
[47] Wang, Y.; Huang, J.; Chen, L., Global conservative solutions of the two-component Camassa-Holm shallow water system, Int. J. Nonlinear Sci., 9, 3, 379-384 (2010) · Zbl 1208.35119
[48] R.T. Glassey, J.K. Hunter, Y. Zheng, Singularities and oscillations in a nonlinear variational wave equation, in: Singularities and Oscillations, IMA 91, in: The IMA Volumes in Mathematics and its Applications book series, Springer, pp. 37-60. · Zbl 0958.35084
[49] Guan, C.; Yin, Z., Global weak solutions for a modified two-component Camassa-Holm equation, Ann. Inst. Henri Poincare C, Non Linear Anal., 28, 4, 623-641 (2011) · Zbl 1241.35159
[50] Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 11, 1661-1664 (1993) · Zbl 0972.35521
[51] Hunter, J. K.; Zheng, Y., On a completely integrable nonlinear hyperbolic variational equation, Physica D, 79, 2-4, 361-386 (1994) · Zbl 0900.35387
[52] Kaye, G. W.C.; Laby, T. H., Tables of physical and chemical constants, Longman Sci. Tech. (1995) · JFM 42.1041.05
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.