A Locally Quadratic Glimm Functional and Sharp Convergence Rate of the Glimm Scheme for Nonlinear Hyperbolic Systems
Tóm tắt
Consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one space dimension
$$u_{t} \, + \, A(u)u_{x} = 0, \qquad u(0, x) = {\bar u}(x), \quad \quad \quad \quad (1)$$
where
$${{u \mapsto A(u)}}$$
is a smooth matrix-valued map, and the initial data
$${{\overline u}}$$
is assumed to have small total variation. We investigate the rate of convergence of approximate solutions of (1) constructed by the Glimm scheme, under the assumption that, letting λ
k
(u), r
k
(u) denote the k-th eigenvalue and a corresponding eigenvector of A(u), respectively, for each k-th characteristic family the linearly degenerate manifold
$$\mathcal{M}_{k} \doteq \left\{u \in \Omega : \nabla\lambda_{k}(u) \cdot r_{k}(u) = 0\right\}$$
is either the whole space, or it is empty, or it consists of a finite number of smooth, N–1-dimensional, connected, manifolds that are transversal to the characteristic vector field r
k
. We introduce a Glimm type functional which is the sum of the cubic interaction potential defined in Bianchini (Discrete Contin Dyn Syst 9:133–166, 2003), and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter. Relying on an adapted wave tracing method, and on the decrease amount of such a functional, we obtain the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems satisfying the classical Lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families.
Tài liệu tham khảo
Ancona F., Marson A.: A note on the Riemann Problem for general n × n conservation laws. J. Math. Anal. Appl. 260, 279–293 (2001)
Ancona F., Marson A. Well-posedness for general 2 × 2 systems of conservation laws. Memoirs Amer. Math. Soc. 169(801) (2004)
Ancona F., Marson A.: A wave front tracking algorithm for N × N non genuinely nonlinear conservation laws. J. Differ. Equ. 177, 454–493 (2001)
Ancona F., Marson A.: Existence theory by front tracking for general nonlinear hyperbolic systems. Arch. Rational Mech. Anal. 185(2), 287–340 (2007)
Bianchini S.: On the Riemann problem for non-conservative hyperbolic systems. Arch. Rational Mech. Anal. 166, 1–26 (2003)
Bianchini S.: Interaction estimates and Glimm functional for general hyperbolic systems. Discrete Contin. Dyn. Syst. 9, 133–166 (2003)
Bianchini S., Bressan A.: Vanishing viscosity solutions to nonlinear hyperbolic systems. Ann. Math. 161, 223–342 (2005)
Bressan A.: Hyperbolic Systems of Conservation Laws – The One-dimensional Cauchy Problem. Oxford University Press, Oxford (2000)
Bressan A., Marson A.: Error bounds for a deterministic version of the Glimm scheme. Arch. Rational Mech. Anal. 142, 155–176 (1998)
Bressan A., Yang T.: On the convergence rate of vanishing viscosity approximations. Comm. Pure Appl. Math. 57, 1075–1109 (2004)
Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag, Berlin (2000)
Glass O., LeFloch P.G.: Nonlinear hyperbolic systems: nondegenerate flux, inner speed variation, and graph solutions. Arch. Rational Mech. Anal. 185(3), 409–480 (2007)
Glimm J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965)
Hua, J., Jiang, Z., Yang, T.: A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws. Conserv. Laws Prepr. Serv. preprint 2008-015 (2008)
Hua J., Yang T.: An improved convergence rate of Glimm scheme for general systems of hyperbolic conservation laws. J. Differ. Equ. 231, 92–107 (2006)
Iguchi T., LeFloch P.G.: Existence theory for hyperbolic systems of conservation laws with general flux-functions. Arch. Rational Mech. Anal. 168, 165–244 (2003)
Lax P.D.: Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566 (1957)
Liu T.P.: The determnistic version of the Glimm scheme. Comm. Math. Phys. 57, 135–148 (1975)
Liu T.P.: The Riemann problem for general 2 × 2 conservation laws. Trans. Am. Math. Soc. 199, 89–112 (1974)
Liu T.P.: The Riemann problem for general systems of conservation laws. J. Differ. Equ. 18, 218–234 (1975)
Liu, T.P.: Admissible solutions of hyperbolic conservation laws. Memoirs Am. Math. Soc. 30(240) (1981)
Liu T.P., Yang T.: Weak solutions of general systems of hyperbolic conservation laws. Comm. Math. Phys. 230, 289–327 (2002)