Nonlinear Problems: Fluid Dynamics
Nonlinear initial-value problems are mostly unexplored. The linear problems in the preceding chapters are generally all special cases of nonlinear ones or become nonlinear when interactions are taken into account. Acoustics becomes fluid dynamics when the amplitudes are finite; Maxwell’s equations and the Dirac equation yield a nonlinear system when the coupling between them is included (see Gross 1966). The new phenomena that appear in nonlinear problems are many and varied. In this chapter, a few of these new phenomena are described, in connection with fluid dynamics. The main conclusion is that some sort of piecewise analytic formulation is needed; the details of such a formulation are likely to remain unclear until much more theoretical work has been done.
KeywordsRelation between linear and nonlinear problems of evolution fluid dynamics as an example system of conservation laws quasilinear equations weak solutions jumps and jump conditions shock slip surface contact discontinuity Rankine-Hugoniot conditions entropy condition characteristics hyperbolic equations characteristic form Riemann invariants Cauchy-Kovalevski theorem noncharacteristic initial surface or initial data characteristic plane the Riemann problem spontaneous generation of shocks Helmholtz and Taylor instabilities piecewise analytic initial-value problem Mach reflection triple shock intersection corner flow power series calculation of detached shock algebraic manipulation of power series by computer significance arithmetic analytic continuation by computer
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