Cauchy initial-value problem

Now, the averaged hyperbolic model, Eq. (52), defines a characteristic initial-value problem (Cauchy problem). To complete the model, we need to specify Cm only along the characteristic curves x = 0 and f — 0. Thus, the initial and boundary conditions for the averaged model are obtained by taking the mixing-cup averages of Eqs. (31) and (32) ... 

This section concerns the Cauchy problem or initial value problem, where initial data at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et al. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a stable region, one could get arbitrarily close to an unstable one. 

To solve for the retardation interactions in the Heisenberg representation, the initial value problem of Cauchy developed in the Minkowski spacetime may be generalized as shown in Appendix 12. A. Now, the symmetric energy-momentnm tensor in Eqnation 12.6 is given as ... 

The solution of Eq. (2.6) for infinite interval and delta-shaped initial distribution (2.8) is called the fundamental solution of Cauchy problem. If the initial value of the Markov process is not fixed, but distributed with the probability density Wo(x), then this probability density should be taken as the initial condition ... 

The existence and uniqueness of the solution of mass-action-type kinetic differential equations (or, more precisely, initial value or Cauchy problems for this type of differential equations) are ensured by general theorems, such as the Picard-Lindelof theorem (see the textbooks cited above). 

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