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It is modern practice to define hyperbolicity of a partial differential operator as a necessary condition for the well-posedness of the Cauchy problem, see [a2] , Vol. Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. The main questions connected with Cauchy problems are as follows: 1 Does there exist albeit only locally a solution? The simplest Cauchy problem is to find a function defined on the half-line , satisfying a first-order ordinary differential equation 1 is a given function and taking a specified value at : 2 In geometrical terms this means that, considering the family of integral curves of equation 1 in the -plane, one wishes to find the curve passing through the point.

For ordinary differential equations of a higher order, the Cauchy problem the initial data of which involve, besides the function itself, the derivatives can be reduced by the standard device to a corresponding problem of type 1 , 2. Then the solvability condition is in a neighbourhood of the given point; here are the symbols for the exterior differential and the exterior product, respectively see Frobenius theorem. For linear partial differential equations 3 the Cauchy problem may be formulated as follows.

The initial conditions may be given in the form of derivatives of with respect to the direction of the unit normal to : 4 where the , , are known functions Cauchy data. The formulation of the Cauchy problem for non-linear differential equations is similar. An illustration is Hadamard's example: The Cauchy problem for the Laplace equation with initial conditions has no solution if is not an analytic function. A typical hyperbolic equation is the wave equation 7 considered in an -dimensional region, with variables.

The Cauchy problem for this equation with data on the hyperplane is uniquely solvable for any sufficiently smooth functions , and the solution depends continuously in some metric on these functions. For the cases and , an explicit form of the solution is given by the formulas of d'Alembert, Poisson and Kirchhoff, respectively: where , ; where , , and is the surface element on the unit sphere.

References [1] S. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint Translated from French [3] L.

### Submission history

Bers, F. John, M. Courant, D. Hilbert, "Methods of mathematical physics. Mizohata, "The theory of partial differential equations" , Cambridge Univ. Tikhonov] Tichonoff, A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. References [a1] P. Treves, "Basic partial differential equations" , Acad.

### Analysis & PDE

Press MR Zbl Encyclopedia of Mathematics. This article was adapted from an original article by A. Publication Date: Hyperbolic Equations. View on repository.

## On C ∞ well-posedness of hyperbolic systems with multiplicities

Pure Mathematics , Cauchy Problem , and Schrodinger equation. Pure Mathematics , Cauchy Problem , and Optimal control hyperbolic partial differential equations. View on worldscinet. Pure Mathematics and Optimal control hyperbolic partial differential equations. Here we consider the general case of a Here we consider the general case of a system with real, possibly multiple, characteristics, and we ask which regularity should be a priori required of a given solution in order that it enjoys the propagation of analyticity.

Applied Mathematics , Pure Mathematics , and Nonlinear system. View on dx. Some algebraic properties of the hyperbolic systems more. The technique of quasi-symmetrizer has been applied to the well-posedness of the Cauchy problem for scalar operators [10], [13] and linear systems [5], [15], [4], and to the propagation of analitycity for solutions to semi-linear systems The technique of quasi-symmetrizer has been applied to the well-posedness of the Cauchy problem for scalar operators [10], [13] and linear systems [5], [15], [4], and to the propagation of analitycity for solutions to semi-linear systems [6].

In all these works, it is assumed that the principal symbol depends only on the time variable. In this note we illustrate, in some special cases, a new property of the quasisymmetrizer which allows us to generalize the result in [6] to semi-linear systems with coefficients depending also on the space variables [21]. Mathematische Annalen. Math Ann. Published online Mar Author information Article notes Copyright and License information Disclaimer.

Corresponding author. Received Jan 10; Revised Mar 8. Abstract In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. Mathematics Subject Classification: 35L45 primary , 46E35 secondary.

Theorem A [ 27 , Theorem 7. Well-posedness in anisotropic Sobolev spaces This section is devoted to proving the well-posedness of the Cauchy problem 1.

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Auxiliary remarks In solving the Cauchy problem 1 , we will deal with solutions of certain scalar pseudo-differential equations. First step or Schur step The first step in our triangularisation follows the construction in the constant case except that we will not get a unitary transformation matrix. The triangularisation procedure The reduction to an upper triangular form or the Schur transformation of A is possible under certain conditions on its eigenvectors. Theorem 7 [ 27 , Theorem 7. Example i. Examples i. Contributor Information Claudia Garetto, Email: ku.

References 1.

Bernstein DS. Matrix Mathematics—Theory, Facts, and Formulas. Princeton: Princeton University Press; Bronshtein MD. Smoothness of roots of polynomials depending on parameters. Colombini F, Kinoshita T. On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order. Colombini F, Spagnolo S. Pisa Cl. Time-dependent loss of derivatives for hyperbolic operators with non regular coefficients.

Partial Differ. A well-posedness result for hyperbolic operators with Zygmund coefficients.

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## On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order

Pures Appl. Nonuniqueness in hyperbolic Cauchy problems. Colombini F, Lerner N. Hyperbolic operators with non-Lipschitz coefficients. Duke Math. Colombini F, Nishitani T. Second order weakly hyperbolic operators with coefficients sum of powers of functions.

Osaka J. On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order. On the 2 by 2 weakly hyperbolic systems. Dieci L, Eirola T. On smooth decompositions of matrices.

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SIAM J.