Fluids Under Pressure

The primitive equations for oceanic and atmospheric dynamics are a fundamental model for many geophysical flows. In this chapter we present a summary of an approach to these equations based on the theory of evolution equations. In particular, we discuss the hydrostatic Stokes operator, well-posedness results in critical spaces within the L ^p(L ^q)-scale, within the scaling invariant space L ^∞(L ¹) for Neumann boundary conditions, and within the L ^∞(L ^p) space for mixed boundary conditions and p > 3. In addition, we investigate real analyticity of solutions, convergence of the scaled Navier-Stokes equations to the primitive equations, and the existence of periodic solutions for large forces.

This chapter is devoted to some mathematical analysis of the two-phase problem for the viscous incompressible–incompressible capillary flows separated by sharp interface, this problem being called two-phase problem for short, and the Stokes equations with transmission conditions on the sharp interface which is arised from the two-phase problem.

The maximal regularity is a character of the system of equations of parabolic type, and it is a very powerful tool in solving quasilinear equations of parabolic type. The authors of this lecture note have developed a systematic method to derive the maximal regularity theorem for the initial-boundary value problem for the Stokes equations with non-homogeneous boundary conditions, which is based on the \(\mathcal {R}\) bounded solution operators theory and L. Weis’ operator valued Fourier multiplier theorem. The notion of \(\mathcal {R}\) boundedness plays an essential role in the Weis’ theory, which takes the place of boundedness in the standard Fourier multiplier theorem of Marcinkiewicz-Mikhilin-Hölmander type.

In this lecture note, we explain how to use the \(\mathcal {R}\)-bounded solution operators to derive the maximal regularity theorem for the Stokes equations with transmission conditions, and as an application of our maximal regularity theorem, we prove the local well-posedness of the two-phase problem, where the solutions are obtained in the L _p in time and L _q in space maximal regularity class. So far, this framework gives us the best possible regularity class of parabolic quasilinear equations.

Moreover, we prove the global well-posedness for the two-phase problem both in the bounded domain case and the unbounded domain case. A key tool is the decay property of the C ⁰ analytic semigroup associated with the Stokes equations with transmission conditions. In the bounded domain case, the decay properties are obtained essentially from the analysis of zero eigenvalue. As a result we prove the exponential stability of our C ⁰ analytic semigroup in some quotient space, which, together with the conservation of momentum and angular momentum and the maximal regularity theorem, yields the global well-posedness in the case of small initial data and the ball-like reference domain.

On the other hand, in the unbounded domain case, the zero is a continuous spectrum for the Stokes equations with transmission conditions, and so we can prove the polynomial decay properties for the C ⁰ analytic semigroup only, which, combined with L _p-L _q maximal regularity theorem with suitable choices of p and q, yields the L _p time summability of the L _q space norm of solutions to the nonlinear problem. From this we prove the global well-posedness for the small initial data in the unbounded domain case. Notice that the L _p summability yields L ₁ summability in view of the Hölder inequality, which is enough to handle the kinetic equations.

What we want to emphasize here is that our method is based on the construction of \(\mathcal {R}\) bounded solution operators and spectral analysis of the zero eigenvalue or generalized eigenvalue of the generalized resolvent problem for the Stokes equations with transmission conditions. The spectral analysis here can be used widely to study the other parabolic linear and quasilinear equations arising from the mathematical study of viscous fluid flows as well as other models in mathematical physics like MHD, multicomponent flows, namatic crystal flows, and so on.