Dissipative structure of one-dimensional isothermal compressible fluids of Korteweg type. (English) Zbl 1504.35454
Summary: This paper studies the dissipative structure of the system of equations that describes the motion of a compressible, isothermal, viscous-capillar fluid of Korteweg type in one space dimension. It is shown that the system satisfies the genuine coupling condition of J. Humpherys [J. Hyperbolic Differ. Equ. 2, No. 4, 963–974 (2005; Zbl 1090.35119)], which is, in turn, an extension to higher order systems of the classical condition by S. Kawashima and Y. Shizuta [Tôhoku Math. J. (2) 40, No. 3, 449–464 (1988; Zbl 0699.35171); Hokkaido Math. J. 14, 249–275 (1985; Zbl 0587.35046)] for second order systems. It is proved that genuine coupling implies the decay of solutions to the linearized system around a constant equilibrium state. For that purpose, the symmetrizability of the Fourier symbol is used in order to construct an appropriate compensating matrix. These linear decay estimates imply the global decay of perturbations to constant equilibrium states as solutions to the full nonlinear system, via a standard continuation argument.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35L65 | Hyperbolic conservation laws |
Keywords:
isothermal Korteweg compressible fluids; genuine coupling; dissipative structure; global decayReferences:
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