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An inviscid regularization of hyperbolic conservation laws. (English) Zbl 1295.35322

Summary: This article examines the utilization of a spatial averaging technique to the nonlinear terms of the partial differential equations as an inviscid shock-regularization of hyperbolic conservation laws. A central motivation is to promote the idea of applying filtering techniques such as the observable divergence method, rather than viscous regularization, as an alternative to the simulation of shocks and turbulence in inviscid flows while, on the other hand, generalizing and unifying previous mathematical and numerical analysis of the method applied to the one-dimensional Burgers’ and Euler equations. This article primarily concerns the mathematical analysis of the technique and examines two fundamental issues. The first is on the global existence and uniqueness of classical solutions for the regularization under the more general setting of quasilinear, symmetric hyperbolic systems in higher dimensions. The second issue examines one-dimensional scalar conservation laws and shows that the inviscid regularization method captures the unique entropy or physically relevant solution of the original, non-averaged problem as filtering vanishes.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L02 First-order hyperbolic equations
35L03 Initial value problems for first-order hyperbolic equations
35L65 Hyperbolic conservation laws
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[1] Bank, M., Ben-Artzi, M.: Scalar conservation laws on a half-line: a parabolic approach. J. Hyperbolic Differ. Equ. 7, 165–189 (2010) · Zbl 1211.35180
[2] Bardos, C., Leroux, A.Y., Nedelec, J.C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979) · Zbl 0418.35024
[3] Bhat, H.S., Fetecau, R.C.: A Hamiltonian regularization of the Burgers equation. J. Nonlinear Sci. 16(6), 615–638 (2006) · Zbl 1108.35107
[4] Bressan, A.: Hyperbolic System of Conservation Laws: The One-Dimensional Cauchy Problem, vol. 20. Oxford University Press, London (2000) · Zbl 0997.35002
[5] Chen, G.Q., Wang, D.: The Cauchy problem for the Euler equations for compressible fluids. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 1. North-Holland, Amsterdam (2002) · Zbl 1230.35096
[6] Chen, S.Y., Holm, D.D., Margoin, L.G., Zhang, R.: Direct numerical simulations of the Navier-Stokes-{\(\alpha\)} model. Physica D 133, 66–83 (1999) · Zbl 1194.76080
[7] Cheskidov, A., Holm, D.D., Olson, E., Titi, E.S.: On a Leray-{\(\alpha\)} model of turbulence. Proc. R. Soc., Math. Phys. Eng. Sci. 461(2055), 629–649 (2005) · Zbl 1145.76386
[8] Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, vol. 325. Springer, Berlin (2010) · Zbl 1196.35001
[9] Dubois, F., Le Floch, P.: Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71(1), 93–122 (1988) · Zbl 0649.35057
[10] Fischer, A.E., Marsden, J.E.: The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I. Commun. Math. Phys. 28, 1–38 (1972) · Zbl 0247.35082
[11] Foias, C., Holm, D.D., Titi, E.S.: The Navier-Stokes-{\(\alpha\)} model of fluid turbulence. Physica D 152(3), 505–519 (2001) · Zbl 1037.76022
[12] Holden, H., Risebro, N.: Front Tracking for Hyperbolic Conservation Laws, vol. 152. Springer, Berlin (2002) · Zbl 1006.35002
[13] Holm, D.D., Marsden, J.E., Ratiu, T.S.: Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 349, 4173–4177 (1998)
[14] Hou, T., Li, C.: On global well-posedness of the Lagrangian averaged Euler equations. SIAM J. Math. Anal. 38, 782–794 (2006) · Zbl 1111.76006
[15] Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58(3), 181–205 (1975) · Zbl 0343.35056
[16] Khouider, B., Titi, E.S.: An inviscid regularization for the surface quasi-geostrophic equation. Commun. Pure Appl. Math. 61(10), 1331–1346 (2008) · Zbl 1149.35018
[17] Kreiss, H.O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–298 (1970) · Zbl 0188.41102
[18] Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’space. Acta Math. 63, 193–248 (1934) · JFM 60.0726.05
[19] Lunasin, E.M., Ilyin, A.A., Titi, E.S.: A modified-Leray-{\(\alpha\)} subgrid scale model of turbulence. Nonlinearity 19(4), 879–897 (2006) · Zbl 1106.35050
[20] Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002) · Zbl 0983.76001
[21] Marsden, J.E., Shkoller, S.: Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-{\(\alpha\)}) equations on bounded domains. Philos. Trans. R. Soc. Lond. A 359(1784), 1449–1468 (2001) · Zbl 1006.35074
[22] McOwen, R.C.: Partial Differential Equations Methods and Applications, vol. 2. Prentice Hall, New York (2003) · Zbl 0849.35001
[23] Mohseni, K.: Derivation of regularized Euler equations from basic principles. In: AIAA Modeling and Simulation Technologies Conference, Chicago, August 10–13, 2009, AIAA paper 2009-5695
[24] Mohseni, K., Kosović, B., Shkoller, S., Marsden, J.E.: Numerical simulations of the Lagrangian averaged Navier-Stokes (LANS-{\(\alpha\)}) equations for homogeneous isotropic turbulence. Phys. Fluids 15(2), 524–544 (2003) · Zbl 1185.76263
[25] Mohseni, K., Zhao, H., Marsden, J.E.: Shock regularization for the Burgers equation. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, January 9–12, 2006, AIAA paper 2006-1516
[26] Norgard, G., Mohseni, K.: A regularization of the Burgers equation using a filtered convective velocity. J. Phys. A, Math. Theor. 41, 1–21 (2008) · Zbl 1222.76017
[27] Norgard, G., Mohseni, K.: On the convergence of convectively filtered Burgers equation to the entropy solution of inviscid Burgers equation. Multiscale Model. Simul. 7(4), 1811–1837 (2009) · Zbl 1422.35137
[28] Norgard, G., 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
[29] Norgard, G., Mohseni, K.: An examination of the homentropic Euler equations with averaged characteristics. J. Differ. Equ. 248, 574–593 (2010) · Zbl 1183.76791
[30] Taylor, M.E.: Partial Differential Equations III: Nonlinear Equations, vol. 117. Springer, Berlin (1996) · Zbl 0869.35004
[31] Zhao, H., Mohseni, K.: A dynamic model for the Lagrangian averaged Navier-Stokes-{\(\alpha\)} equations. Phys. Fluids 17(7), 075106 (2005) · Zbl 1187.76597
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