Notes on Hyperbolic Conservation Laws (original) (raw)

Lecture notes on hyperbolic conservation laws

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of the conservation equations and definition of weak solutions. 2. Shocks, Rankine-Hugoniot equations and admissibility conditions. 3. Genuinely nonlinear and linearly degenerate characteristic fields. Solution to the Riemann problem. Wave interaction estimates. 4. Weak solutions to the Cauchy problem, with initial data having small total variation. Proof of global existence via front-tracking approximations. 5. Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. 6. Uniqueness of entropy-admissible weak solutions. 7. Approximate solutions constructed by the Glimm scheme. 8. Vanishing viscosity approximations. 9. Counter-examples to global existence, uniqueness, and continuous dependence of solutions, when some key hypotheses are removed. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous chapters.

Hyperbolic Conservation Laws: An Illustrated Tutorial

Lecture Notes in Mathematics, 2012

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: (1) Meaning of a conservation equation and definition of weak solutions. (2) Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. (3) Shock waves: Rankine-Hugoniot equations and admissibility conditions. (4) Genuinely nonlinear and linearly degenerate characteristic fields. Centered rarefaction waves. The general solution of the Riemann problem. Wave interaction estimates. (5) Weak solutions to the Cauchy problem, with initial data having small total variation. Approximations generated by the front-tracking method and by the Glimm scheme. (6) Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. (7) Characterization of solutions which are limits of front tracking approximations. Uniqueness of entropy-admissible weak solutions. (8) Vanishing viscosity approximations. (9) Extensions and open problems. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous sections. Throughout the exposition, technical details are mostly left out. The main goal of these notes is to convey basic ideas, also with the aid of a large number of figures.

Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws

Archive for Rational Mechanics and Analysis, 2000

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ∞ as t→0+ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation.

Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics

Journal of Differential Equations, 1981

The constraints under which a gas at a certain state will evolve can be given by three partial differential equations which express the conservation of momentum, mass, and energy. In these equations, a particular gas is defined by specifying the constitutive relation e = e(u, S), where e = specific internal energy, v = specific volume, and S = specific entropy. The energy function e = -In u + (S/R) describes a polytropic gas for the exponent y = 1, and for this choice of e(V, S), global weak solutions for bounded measurable data having finite total variation were given by Nishida in [lo]. Here the following general existence theorem is obtained: let e,(v, S) be any smooth one parameter family of energy functions such that at E = 0 the energy is given by e&v, S) = -In v + (S/R). It is proven that there exists a constant C independent of E, such that, if E . (total variation of the initial data) < C, then there exists a global weak solution to the equations. Since any energy function can be connected to e&V, S) by a smooth parameterization, our results give an existence theorem for all the conservation laws of gas dynamics. As a corollary we obtain an existence theorem of Liu, Indiana Univ. Math. J. 26, No. 1 (1977) for polytropic gases. The main point in this argument is that the nonlinear functional used to make the Glimm Scheme converge, depends only on properties of the equations at E = 0. For general n x n systems of conservation laws, this technique provides an alternate proof for the interaction estimates in Glimm's 1965 paper. The new result here is that certain interaction differences are bounded by E as well as by the approaching waves.

Existence of Solutions to Hyperbolic Conservation Laws with a Source

Communications in Mathematical Physics, 1997

Existence of solutions to three different systems of Eqs. (1.1), (1.2) and (1.3) coming from physically relevant models is shown, each needing a different proof which are given in Sects. 2, 3 and 4. The unifying theme is the presence of source terms and the general method of proof is vanishing viscosity together with compensated compactness. For system (1.2) entropy-entropy flux pairs of Lax type are constructed and estimates from singular perturbation theory of ODEs are used. For (1.1) and (1.3) weak entropyentropy flux pairs are constructed following the compensated compactness framework set up by Diperna [4].

Hyperbolic systems of conservation laws

1991

Titis ja a survey paper, written iii the occasion of an invited tahk given by tite autitor at tbe Universidad Complutense in Madrid, Octoher 1998. Its purpose is to provide an account of sorne recent advances in tite matitematical theory of byperbolic systems of conservation laws in one apace dimension. After a brief review of basic concepts, we describe in detail tbe metitod of wave-front tracking approximation and present sorne of tbe latest resulta on uniquenees and stabiity of entropy weak sohutions.

Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws

Indiana University Mathematics Journal, 1999

The paper is concerned with the qualitative structure of entropy solutions to a strictly hyperbolic, genuinely nonlinear system of conservation laws. We first give an accurate description of the local and global wave-front structure of a BV solution, generated by a front tracking algorithm. We then consider a sequence of exact or approximate solutions u ν , converging to a solution u in L 1. The convergence of the wave-fronts of u ν to the corresponding fronts of u is studied, proving a structural stability result in a neighborhood of each point in the t-x plane.