This paper addresses the weak-strong uniqueness property and singular limit for the compressible Primitive Equations (PE). We show that a weak solution coincides with the strong solution emanating from the same initial data. On the other hand, we prove compressible PE will approach the incompressible inviscid PE equations in the regime of low Mach number and large Reynolds number in the case of well-prepared initial data. To the best of the authors' knowledge, this is the first work to bridge the link between the compressible PE with incompressible inviscid PE.

In this paper, we study a compressible Primitive Equations (CPEs) of the ocean in the two dimensional space, with horizontal periodic and vertical mixed boundary conditions. Thanks to an effective change of variables, we obtain a new CPEs model, which is similar as viscous shallow water equation. Using a new entropy estimate, we prove the stability of weak solutions for this new two dimensional CPEs model.

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier-slip condition. The Navier-slip condition differs in many aspects from no-slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax-Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier-slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L 1 norm of the deformation tensor D.u/.

In this paper, we consider compressible Navier-Stokes-Korteweg (N-S-K) equations with more general pressure laws, that is the pressure P is non-monotone. We prove the stability of weak solutions in the periodic domain Ω = T N , when N = 2, 3. Utilizing an interesting Sobolev inequality to tackle the complicated Korteweg term, we obtain the global existence of weak solutions in one dimensional case. Moreover, when the initial data is compactly supported in the whole space R, we prove the compressible N-S-K equations will blow-up in finite time.

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