Abstract
This paper concerns the study of instability of a generic compressible two-fluid model in the whole space , where the capillary pressure is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, , Evje et al (2016 Arch. Ration. Mech. Anal. 221 1285–316) established global stability of the constant equilibrium state for the three-dimensional Cauchy problem under some smallness assumptions. Recently, Wu et al (2022 arXiv:2204.10706) proved global stability of the constant equilibrium state for the case (corresponding to ). We investigate instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, . By employing the Hodge decomposition technique and making a detailed analysis of the Green’s function for the corresponding linearized system, we construct solutions of the linearized problem that grow exponentially in time in the Sobolev space Hk , thus leading to a global instability result for the linearized problem. Moreover, with the help of this global linear instability result together with a local existence theorem of classical solutions to the original nonlinear system, we can show instability of the nonlinear problem in the sense of Hadamard by carefully analysing the properties of the semigroup. Therefore, our results show that for the case , the constant equilibrium state of the two-fluid model is globally linearly unstable and locally nonlinearly unstable in the sense of Hadamard, which is different from the cases (Evje et al 2016 Arch. Ration. Mech. Anal. 221 1285–316) and (corresponding to ) (Wu et al 2022 arXiv:2204.10706) where the constant equilibrium state of the two-fluid model was proved to be globally nonlinearly stable.
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