Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

A. A. Zlotnik   ; B. N. Chetverushkin

Sbornik. Mathematics, Tome 214 (2023) no. 4, pp. 444-478

Résumé

The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $\tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(\tau^{\alpha/2})$ in the norm of $C(0,T;L^2(\mathbb{R}^n))$, for an initial function $\mathbf w_0$ in the Sobolev space $H^\alpha(\mathbb{R}^n)$, $\alpha=1,2$, or the Nikolskii space $H_2^{\alpha}(\mathbb{R}^n)$, $0<\alpha<2$, $\alpha\neq 1$, and under appropriate assumptions on the free term of the first-order system. For ${\alpha=1/2}$ a wide class of discontinuous functions $\mathbf w_0$ is covered. Estimates for derivatives of any order with respect to $x$ for solutions and of order $O(\tau^{\alpha/2})$ for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented. Bibliography: 34 titles.

Détail

Keywords: linear systems of partial differential equations, small parameter, linearized system of gas dynamic equations, quasi-gasdynamic systems of equations, estimates of the difference of solutions.

Bibliographie

[1] B. N. Chetverushkin, Kinetic schemes and quasi-gasdynamic system of equations, MAKS Press, Moscow, 2004, 328 pp. ; English transl., CIMNE, Barcelona, 2008, 298 pp. | Zbl

[2] T. G. Elizarova, Quasi-gasdynamic equations and methods for computation of viscous flows, Nauchnyi Mir, Moscow, 2007, 349 pp. (Russian); English transl., Quasi-gas dynamic equations, Springer, Berlin, 2008, xiv+286 pp. | DOI

[3] B. N. Chetverushkin, “Hyperbolic quasi-gasdynamic system”, Mat. Model., 30:2 (2018), 81–98 ; English transl. in Math. Models Comput. Simul., 10:5 (2018), 588–600 | MR | Zbl | DOI

[4] L. C. Evans, Partial differential equations, Grad. Stud. Math., 19, Amer. Math. Soc., Providence, RI, 1998, xviii+662 pp. | DOI | MR | Zbl

[5] J. D. Cole, Perturbation methods in applied mathematics, Blaisdell Publishing Co. [Ginn and Co.], Waltham, MA–Toronto, ON–London, 1968, vi+260 pp. | MR | Zbl

[6] J. Genet and M. Madaune, “Singular perturbations for a class of nonlinear hyperbolic-hyperbolic problems”, J. Math. Anal. Appl., 64:1 (1978), 1–24 | DOI | MR | Zbl

[7] L. R. Volevich and M. G. Dzhavadov, “Uniform bounds of solutions of hyperbolic equations with a small parameter multiplying the leading derivatives”, Differ. Uravn., 19:12 (1983), 2082–2090 ; English transl. in Differ. Equ., 19 (1983), 1516–1525 | MR | Zbl

[8] A. van Harten and R. R. van Hassel, “A quasi-linear, singular perturbation problem of hyperbolic type”, SIAM J. Math. Anal., 16:6 (1985), 1258–1267 | DOI | MR | Zbl

[9] S. Schochet, “Hyperbolic-hyperbolic singular limits”, Comm. Partial Differential Equations, 12:6 (1987), 589–632 | DOI | MR | Zbl

[10] H. O. Fattorini, “The hyperbolic singular perturbation problem: an operator theoretic approach”, J. Differential Equations, 70:1 (1987), 1–41 | DOI | MR | Zbl

[11] E. M. de Jager and F. Jiang, The theory of singular perturbations, North-Holland Ser. Appl. Math. Mech., 42, North-Holland Publishing Co., Amsterdam, 1996, xii+340 pp. | MR

[12] S. Mizohata, Henbidun hôteisiki ron [The theory of partial differential equations], Contemp. Math., 9, Iwanami Shoten, Tokyo, 1965, viii+462 pp. | MR

[13] S. K. Godunov, Equations of mathematical physics, 2nd revised and augmented ed., Nauka, Moscow, 1979, 391 pp. (Russian) ; French transl. of 1st ed., S. Godounov, Équations de la physique mathématique, Éditions Mir, Moscow, 1973, 452 pp. | MR | MR | Zbl

[14] S. Benzoni-Gavage and D. Serre, Multidimensional hyperbolic partial differential equations. First-order systems and applications, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press, Oxford, 2007, xxvi+508 pp. | MR | Zbl

[15] S. M. Nikol'skiĭ, Approximation of functions of several variables and imbedding theorems, Nauka, Moscow, 1969, 480 pp. ; English transl., Grundlehren Math. Wiss., 205, Springer-Verlag, New York–Heidelberg, 1975, viii+418 pp. | MR | Zbl | DOI | MR | Zbl

[16] A. A. Zlotnik and B. N. Chetverushkin, “Parabolicity of the quasi-gasdynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them”, Zh. Vychisl. Mat. Mat. Fiz., 48:3 (2008), 445–472 ; English transl. in Comput. Math. Math. Phys., 48:3 (2008), 420–446 | MR | Zbl | DOI

[17] A. A. Zlotnik and B. N. Chetverushkin, “Stability of implicit difference schemes for a linearized hyperbolic quasi-gasdynamic system of equations”, Differ. Uravn., 56:7 (2020), 936–947 ; English transl. in Differ. Equ., 56:7 (2020), 910–922 | DOI | MR | Zbl | DOI

[18] A. A. Zlotnik and B. N. Chetverushkin, “On second-order parabolic and hyperbolic perturbations of a first-order hyperbolic system”, Dokl. Ross. Akad. Nauk. Mat. Inform. Protsessy Upr., 506 (2022), 9–15 ; English transl. in Dokl. Math., 106:2 (2022), 308–314 | DOI | Zbl | DOI

[19] H. O. Fattorini, “Singular perturbation and boundary layer for an abstract Cauchy problem”, J. Math. Anal. Appl., 97:2 (1983), 529–571 | DOI | MR | Zbl

[20] A. Z. Ishmukhametov, “Controllability of hyperbolic systems under singular perturbations”, Differ. Uravn., 36:2 (2000), 241–250 ; English transl. in Differ. Equ., 36:2 (2000), 272–284 | MR | Zbl | DOI

[21] T. E. Moiseev, E. E. Myshetskaya and V. F. Tishkin, “On the closeness of solutions of unperturbed and hyperbolized heat equations with discontinuous initial data”, Dokl. Ross. Akad. Nauk, 481:6 (2018), 605–609 ; English transl. in Dokl. Math., 98:1 (2018), 391–395 | DOI | Zbl | DOI | MR

[22] B. N. Chetverushkin and A. A. Zlotnik, “On a hyperbolic perturbation of a parabolic initial-boundary value problem”, Appl. Math. Lett., 83 (2018), 116–122 | DOI | MR | Zbl

[23] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva (Ural'tseva), Linear and quasi-linear equations of parabolic type, Nauka, Moscow, 1967, 736 pp. ; English transl., Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968, xi+648 pp. | MR | Zbl | DOI | MR | Zbl

[24] O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Nauka, Moscow, 1973, 407 pp. ; English transl., Appl. Math. Sci., 49, Springer-Verlag, New York, 1985, xxx+322 pp. | MR | Zbl | DOI | MR | Zbl

[25] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Textb. Math., 2nd rev. ed., CRC Press, Boca Raton, FL, 2015, xiv+299 pp. | MR | Zbl

[26] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Math. Lehrbucher und Monogr., 38, Akademie-Verlag, Berlin, 1974, ix+281 pp. | MR | Zbl

[27] A. A. Zlotnik and B. N. Chetverushkin, “Spectral stability conditions for an explicit three-level finite-difference scheme for a multidimensional transport equation with perturbations”, Differ. Uravn., 57:7 (2021), 922–931 ; English transl. in Differ. Equ., 57:7 (2021), 891–900 | DOI | MR | Zbl | DOI

[28] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren Math. Wiss., 223, Springer-Verlag, Berlin–New York, 1976, x+207 pp. | DOI | MR | Zbl

[29] B. N. Chetverushkin and A. A. Zlotnik, “On some properties of multidimensional hyperbolic quasi-gasdynamic systems of equations”, Russ. J. Math. Phys., 24:3 (2017), 299–309 | DOI | MR | Zbl

[30] A. A. Zlotnik, Projection-difference methods for nonstationary problems with nonsmooth data, Kandidat Dissertation, Moscow State University, Moscow, 1979 (Russian)

[31] L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lect. Notes Unione Mat. Ital., 3, Springer, Berlin; UMI, Bologna, 2007, xxvi+218 pp. | DOI | MR | Zbl

[32] M. S. Agranovich, Sobolev spaces, their generalizations and elliptic problems in smooth and Lipschitz domains, Moscow Center of Continuous Mathematical Education, Moscow, 2013, 378 pp.; English transl., Springer Monogr. Math., Springer, Cham, 2015, xiii+331 pp. | DOI | MR | Zbl

[33] A. Zlotnik and T. Lomonosov, “$L^2$-dissipativity of the linearized explicit finite-difference scheme with a kinetic regularization for 2D and 3D gas dynamics system of equations”, Appl. Math. Lett., 103 (2020), 106198, 7 pp. | DOI | MR | Zbl

[34] A. A. Zlotnik and A. S. Fedchenko, “Properties of an aggregated quasi-gasdynamic system of equations for a homogeneous gas mixture”, Dokl. Ross. Akad. Nauk Mat. Inform. Protsessy Upr., 501 (2021), 31–37 ; English transl. in Dokl. Math., 104:3 (2021), 340–346 | DOI | Zbl | DOI | MR