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2 Constant-Coefficient Cauchy Problems

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)62304-1


Publisher Summary This chapter discusses the constant-coefficient Cauchy problems. The main tool to discuss constant-coefficient problems is the Fourier transformation. It allows decomposing general initial data into pure exponentials. In case of constant coefficients, the time evolution of each pure (spatial) exponential can be treated separately. The chapter focuses on hyperbolic and parabolic systems in one space dimension. It characterizes families of matrices A for which the exponential is uniformly bounded. The result that is central in a general theory of well-posedness is applied to characterize those first-order systems (in any number of space dimensions), which lead to well-posed Cauchy problems; the corresponding systems are called “strongly hyperbolic.” An important example is presented, the compressible Euler equations linearized about a constant flow. Similarly, linearization about a constant flow of the viscous compressible Navier-Stokes equations leads to a mixed hyperbolic-parabolic system; applying the same general principles, one can obtain well-posedness of the Cauchy problem and can solve these linearized problems by Fourier transformation.

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