Abstract. A modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation. We provide a mathematical formulation of the problem that appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete theory for the solutions to what turns out to be a nonlinearly coupled system of non-smooth partial differential equations. Specifically we prove global existence of quasi-strong and strong solutions, along with uniqueness results and maximum principles of physical interest.

We consider the linearized 2D inviscid shallow water equations in a rectangle. A set of boundary conditions is proposed which make these equations well-posed. Several different cases occur depending on the relative values of the reference velocities (u0, v0) and reference height φ0 (sub-or super-critical flow at each part of the boundary).

Abstract. In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem.

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We address the well-posedness of the 2D (Euler)-Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich's type data, which gives a positive answer to part of the questions raised in [LPZ11]. Our analysis on the the polygonal-like domains essentially relies on the recent elliptic regularity results for such domains proved in [BDT13, DT13].Euler-Boussinesq equations describing the evolution of mass and heat flow of the inviscid incompressible fluid read in the non-dimensional form:where (x, y) ∈ Ω, t ∈ (0, t 1 ), u = (u 1 , u 2 ) and T denote the velocity field and the temperature of the fluid respectively, and π stands for the pressure and κ > 0 is the thermal diffusivity. We associate with (1.1) the following initial and boundary conditions:where n is the outward unit normal vector to ∂Ω and u 0 , T 0 and η are the given initial and boundary data. We also denote by τ the unit tangent vector to ∂Ω.The general 2D Boussinesq system with full viscosity ν and diffusivity κ readsFrom the mathematical point of view, the global well-posedness, global regularity as well as the existence of the global attractor of the Boussinesq system have been widely studied, see for example [CD80, FMT87, Guo89, ES94, CN97, MZ97, Wang05, CLR06, Wang07, Xu09, KTW11, CW12]. Recently, there are many works devoted to the study of the 2D Boussinesq system with partial viscosity, see for example [HL05, Cha06, HK07, DP09, HK09, HKR11] in the whole space R 2 and [Zha10, LPZ11, HKZ13] in bounded smooth domains. There are also many works which considered the case when only the horizontal viscosity and vertical viscosity is present, see for example [ACW11, DP11, CW13, MZ13]. However, the global regularity for the 2D Boussinesq system when ν = κ = 0 is still an outstanding open problem and to the best of our knowledge, the well-posedness issue regarding the 2D Euler-Boussinesq system (1.1) in non-smooth domains has not yet been addressed in the literatures, which is the goal of this article. In some realistic applications, the variation of the fluid viscosity and thermal diffusivity with the temperature may not be disregarded (see for example [LB96] and references therein) and there are many works on this direction, too, see for example [LB96,LB99,SZ13,LPZ13,Hua14] where the existence of weak solutions, global regularity, and existence of global attractor have been studied. It is well known that the standard 2D Euler equations is globally well-posedness if the initial data satisfies Yudovich's type condition, see [Yud63,Yud95,Kel11]. Roughly speaking, if the initial vorticity is bounded or unbounded but with small growth rate of the L p -norm, then the 2D Euler equations exist a global unique solution and recently this result has been extended to non-smooth domains in [BDT13,DT13]. Note that the global well-posedness for the 2D Euler-Boussinesq system has been studied in [DP09] with Yudovich's type data for the whole space R 2 and also studied in [Zha10] with H 3 -regular data for bounded smooth

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichssymmetrizable.

Abstract. Global well-posedness of strong solutions and existence of the global attractor to the initial and boundary value problem of 2D Boussinesq system in a periodic channel with non-homogeneous boundary conditions for the temperature and viscosity and thermal diffusivity depending on the temperature are proved.

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