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Existence of solutions for compressible fluid models of Korteweg type. (English) Zbl 1010.76075

The authors prove existence and uniqueness of smooth solutions for an isothermal model of capillary compressible fluids derived by J. E. Dunn and J. Serin [Arch. Ration. Mech. Anal. 89, 95-133 (1985; Zbl 0582.73004)]. This model can be also used as a simplified model of phase transition. First, the well-posedness in critical Besov spaces is studied. The authors prove global existence and uniqueness of solutions close to a stable equilibrium, and then the local in time existence for solutions with pressure law allowing spinodal regions. Assuming a lower and upper control of the density, the authors also show the existence of weak solutions in two-dimensional case near equilibrium. For non-capillary case, some blow-up properties are described.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 0582.73004
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References:

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