Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems

By Stefano Bianchini, Alberto Bressan. 2001

551 citations recorded by the source corpus.

We consider the Cauchy problem for a strictly hyperbolic, n × n system in one-space dimension: ut + A(u)ux = 0, assuming that the initial data have small total variation. We show that the solutions of the viscous approximations ut + A(u)ux = euxx are defined globally in time and satisfy uniform BV estimates, independent of e. Moreover, they depend continuously on the initial data in the L 1 distance, with a Lipschitz constant independent of t, e. Letting e → 0, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where A = Df is the Jacobian

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17b27112e8f4f114a702d9f544eb0b8537b3597d

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Physics-informed neural networks for inverse problems in supersonic flows (2022, background): “The viscosity solutions [33] of conservation laws are associated with the zero-viscosity limit given by U = lim (cid:15) → 0 + U (cid:15) , where ∂ t U (cid:15) + ∇ · G ( U (cid:15) ) = (cid:15) ∇ 2 U (cid:15) .”
Statistical Solutions of Hyperbolic Conservation Laws: Foundations (2016, unknown): “For one-dimensional systems ( d = 1, N > 1 in (1.1)), global existence, under the assumption of small initial total variation, was shown by Glimm in [32] and by Bianchini and Bressan in [6].”
The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems (2011, methodology): “The well-posedness theory for these equations is presently understood only in the scalar case k = 1, and this goes back to Kružkov [103], and in the one-dimensional case d = 1, via the Glimm scheme [97], or the front tracking method of Dafermos [74], or the more recent vanishing viscosity method of ”
Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields (2009, methodology): “The well-posedness theory for these equations is presently understood only in the scalar case k = 1, and this goes back to Kružkov [103], and in the one-dimensional case d = 1, via the Glimm scheme [97], or the front tracking method of Dafermos [74], or the more recent vanishing viscosity method of ”
Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow (2009, methodology): “The vanishing artificial viscosity limit to general strictly hyperbolic systems of conservation laws with general small BV initial data was first established by Bianchini-Bressan [3] via direct BV estimates with small oscillation.”
Numerical methods for nonconservative hyperbolic systems: a theoretical framework (2006, background): “Many papers have been devoted to the definition and stability of nonconservative products, and its application to the definition of weak solutions of nonconservative hyperbolic systems; see [1], [4], [6], [9], [13], [14], [26], [34], [41].”
Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods (2008, unknown): “Many papers have been devoted to the definition and the stability of non-conservative products and its application to the definition of weak solutions of non-conservative hyperbolic systems: see [63], [11], [15], [16], [48], [7], [38], [5], [1] .”
On the Riemann Problem for Non-Conservative Hyperbolic Systems (2003, methodology): “We recall that in [ 3 ] the curve Ti is proved to be Lipschitz.”
Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system (2010, methodology): “For the well-posedness problems, we refer to the works [5,6,24,29] (and the references therein) in the content of weak solutions to systems (including scalar case) in conservation laws, and [26,28] in the content of classical solutions to general quasi-linear hyperbolic systems.”