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Abstract: |
We are concerned with stability and instability of the steady state (1,0, 1, 0) for a generic non–conservative compressible two–fluid model in R^3. Under the assumption that the initial fraction densities are close to the constant state (1,1) in H^3\cap\dot B^{s}_{1,\infty} and the initial velocities are small in H^2\cap\dot B^{s}_{1,\infty} with s\in [0,1], it is shown that \frac{1}{2} is the critical value of s on the stability of the model in question. More precisely, when 0\leq s<\frac{1}{2}, the steady state (1,0, 1, 0) is nonlinearly globally stable; and conversely, the steady state (1,0, 1, 0) is nonlinearly unstable in the sense of Hadamard when \frac{1}{2}<s\le 1. Furthermore, for the critical case s=\frac{1}{2}, if the initial data satisfy additional regularity assumption, then the steady state (1,0, 1, 0) is nonlinearly globally stable. |
