Abstract
A careful analysis of the single quasi-linear second-order equation is the gateway into the world of higher-order partial differential equations and systems. One of the most important aspects of this analysis is the distinction between hyperbolic, parabolic and elliptic types. From the physical standpoint, the hyperbolic type corresponds to physical systems that can transmit sharp signals over finite distances.
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Notes
- 1.
The theorem of existence and uniqueness, on the other hand, requires much less than analyticity.
- 2.
For a detailed proof, see the classical treatise [1].
- 3.
In some textbooks, the classification is based on the so-called normal forms of the equations. In order to appreciate the meaning of these forms, however, it is necessary to have already seen an example of each. We prefer to classify the equations in terms of their different behaviour vis-Ã -vis the Cauchy problem. Our treatment is based on [4], whose clarity and conciseness are difficult to match.
- 4.
Although we have assumed that the first coefficient of the PDE does not vanish, in fact the conclusion that the number of characteristic directions is governed by the discriminant of the quadratic equation is valid for any values of the coefficients, provided, of course, that not all three vanish simultaneously.
- 5.
- 6.
In using this terminology, we are implicitly assuming that we have defined the natural dot product in \({\mathbb R}^n\). A more delicate treatment, would consider the gradient not as a vector but as a differential form which would then be annihilated by vectors forming a basis on the singular surface. We have already discussed a similar situation in Box 3.2.
- 7.
For the treatment of the quasi-linear case, see Box 6.2.
- 8.
The proof is not as straightforward as it may appear from the casual reading of some texts. A good treatment of these normal or canonical forms can be found in [2].
References
Courant R, Hilbert D (1962) Methods of mathematical physics, vol 2. Interscience, Wiley, New York
Garabedian PR (1964) Partial differential equations. Wiley, New York
Hadamard J (1903) Leçons sur la Propagation des Ondes et les Équations de l’Hydrodynamique. Hermann, Paris. www.archive.org
John F (1982) Partial differential equations. Springer, Berlin
Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S (ed), Handbuch der Physik. Springer, Berlin
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Epstein, M. (2017). The Second-Order Quasi-linear Equation. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_6
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DOI: https://doi.org/10.1007/978-3-319-55212-5_6
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