Abstract
A first-order quasi-linear p.d.e. is an expression of the form
where x ranges over a region Ω ⊂ R N, the function u: Ω → R is of class C 1, and
are given smooth functions of their arguments.
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References
J.M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Acad. Sci. Amsterdam, Vol. 43 (1940), pp. 2–12; J.M. Burgers, A mathematical model illustrating the theory of turbulence, in Advances in Applied Mechanics, ed. R. von Mises and T. von Kdrmân, Vol. 1, 1948, pp. 171–199.
This is a special case of the Rankine—Hugoniot shock condition. William John Macqorn Rankine, 1820–1872; W.J.M. Rankine, On the thermodynamic theory of waves of finite longitudinal disturbance, Trans. Royal Soc. of London, Vol. 160 (1870), pp. 277–288. Pierre Henri Hugoniot, 1851–1887; H. Hugoniot, Sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits, Journ. de l’École Polytechnique, Vol. 58 (1889), pp. 1–125.
The notion of shock will be made more precise in Section 14.
To justify the terminology, see the derivation of the equation of continuity in Section 2 of the Preliminaries.
A similar concept occurs in the context of the heat equation. See Chapter V, Section 6.
See 4.13 of the Complements of Chapter II and Lemma 6.1c of the Complements of Chapter IV.
The existence of solutions to the Burgers equation was established by Eberhard Hopf, 1902–1983; E. Hopf, The partial differential equation u, + uu x = hxx, Comm. Pure Appl. Math. #3 (1950), pp. 201–230. The variational method we present here, as well as the notion of entropy solutions are taken from P. Lax, Non-linear hyperbolic equations, Comm. Pure Appl. Math. #6 (1953), pp. 231–258. P. Lax, On the Cauchy problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math. #9 (1956), pp. 135–169.
The function U and this Cauchy problem arise naturally from the viscosity method. See Section 7 of the Complements.
The function n(·, t) can be defined by selecting for each x E R an arbitrary element out of the set H(x).
See Section 8 of the Complements.
Compare with (7.8c) of the Complements.
See 11.1 of the Complements.
Compare with 6.1–6.3 of the Complements.
Since, in general, u(·, t) ¢ L I (R), the quantity M (t) is meant in the sense of improper integrals, provided it exists.
Of these only (6.10) is an entropy solution. This notion is introduced and discussed in Section 14.
See also the explicit solutions of Section 6.1 of the Complements. We stress that Theorem 16.1 holds for the solution constructed in Sections 8–10. These are viscosity solutions in the sense specified in the next section. The solution in (6.9) to the problem (6.8) is not a viscosity solution and does not satisfy (16.3).
S.N. Kruzhkov, First order quasi-linear equations in several independent variables, Mat. USSR Sbornik, Vol. 10, #2 (1970), pp. 217–243.
For a formal derivation and a motivation of such a notion, see Section 17 of the Complements. The first notion of entropy solution is due to Lax for equations in one space dimension (see footnote 7). A more general notion, that would cover some cases of nonconvex F(.), and would insure stability, still in one space dimension, is due to O.A. O. einik, Discontinuous solutions of non-linear differential equations, Uspekhi Mat. Nauk (N.S.), Vol. 12 #3 (1957), pp. 3–73 (Amer. Math. Soc. Transl., Ser. 2, 26, pp. 95–172). O.A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation, Uspekhi Mat. Nauk (N.S.), Vol. 14 #6 (1959), pp. 165–170.
See Section 4.13 of the Complements of Chapter II.
See Section 6 of the Complements in the Preliminaries.
The requirement (21.3) does not follow from (21.2). As an example one might take.
See footnote 7.
The variable x will vary according to the definition of i.
For the Burgers equation, the constant L appearing in (16.3) of Theorem 16.1 is one.
The remarks in this section are due to H. Hopf. See footnote 7.
See (2.7) and Theorem 2.1 of Chapter V.
For some cases of nonconvex F(·) see A.S. Kalashnikov, Construction of generalized solutions of quasi-linear equations of first order without convexity conditions as limits of solutions of parabolic equations with a small parameter, Dokl. Akad. Nauk SSSR, 127 (1959), pp. 27–30.
By Problem 3.2 of the Complements of Chapter V, the presence of the term u„ u, is immaterial for a maximum principle to hold.
See the paper of Kruzhkov cited in footnote 17. See also A.I. Vol’ pert, The spaces BV and quasi-linear equations, Math. USSR Sbornik,#2 (1967), pp. 225–267. For a systematic theory of functions of bounded variation, see the monograph of Giusti, [13].
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DiBenedetto, E. (1995). Equations of First Order and Conservation Laws. In: Partial Differential Equations. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-2840-5_8
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