The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case

The primitive equations (PEs) of the atmosphere and the ocean without viscosity are considered. A 2.5D model is introduced, whose motivation is described in the Introduction. A set of nonlocal boundary conditions is proposed, and well-posedness is established for the flows linearized around a constant velocity stratified flow; homogeneous and nonhomogeneous boundary conditions are considered. A related model of dimension 2.5, of physical interest but with fewer degrees of freedom, is also considered at the end.

Download Full-text

Global Well-Posedness of the Ocean Primitive Equations with Nonlinear Thermodynamics

Journal of Mathematical Fluid Mechanics ◽

10.1007/s00021-021-00596-w ◽

2021 ◽

Vol 23 (3) ◽

Author(s):

Peter Korn

Keyword(s):

Equation Of State ◽

Equations Of State ◽

Boussinesq Equations ◽

Climate Science ◽

Rational Functions ◽

Ocean Dynamics ◽

Global Ocean ◽

Primitive Equations ◽

Well Posedness ◽

Nonlinear Thermodynamics

AbstractWe consider the hydrostatic Boussinesq equations of global ocean dynamics, also known as the “primitive equations”, coupled to advection–diffusion equations for temperature and salt. The system of equations is closed by an equation of state that expresses density as a function of temperature, salinity and pressure. The equation of state TEOS-10, the official description of seawater and ice properties in marine science of the Intergovernmental Oceanographic Commission, is the most accurate equations of state with respect to ocean observation and rests on the firm theoretical foundation of the Gibbs formalism of thermodynamics. We study several specifications of the TEOS-10 equation of state that comply with the assumption underlying the primitive equations. These equations of state take the form of high-order polynomials or rational functions of temperature, salinity and pressure. The ocean primitive equations with a nonlinear equation of state describe richer dynamical phenomena than the system with a linear equation of state. We prove well-posedness for the ocean primitive equations with nonlinear thermodynamics in the Sobolev space $${{\mathcal {H}}^{1}}$$ H 1 . The proof rests upon the fundamental work of Cao and Titi (Ann. Math. 166:245–267, 2007) and also on the results of Kukavica and Ziane (Nonlinearity 20:2739–2753, 2007). Alternative and older nonlinear equations of state are also considered. Our results narrow the gap between the mathematical analysis of the ocean primitive equations and the equations underlying numerical ocean models used in ocean and climate science.

Download Full-text

Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽

10.3390/math9050461 ◽

2021 ◽

Vol 9 (5) ◽

pp. 461

Author(s):

Kenta Oishi ◽

Yoshihiro Shibata

Keyword(s):

Free Surface ◽

Free Boundary ◽

Boundary Conditions ◽

Stokes Equations ◽

Mhd Flow ◽

Transmission Conditions ◽

Well Posedness ◽

Anisotropic Space ◽

Free Boundary Conditions ◽

Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

Download Full-text

On well-posedness of source identification elliptic problem with nonlocal boundary conditions

10.1063/5.0040281 ◽

2021 ◽

Author(s):

Allaberen Ashyralyev ◽

Charyyar Ashyralyyev ◽

Viktor G. Zvyagin

Keyword(s):

Boundary Conditions ◽

Elliptic Problem ◽

Source Identification ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Well Posedness

Download Full-text

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

Asymptotic Analysis ◽

10.3233/asy-211703 ◽

2021 ◽

pp. 1-27

Author(s):

Ahmad Makki ◽

Alain Miranville ◽

Madalina Petcu

Keyword(s):

Fractal Dimension ◽

Boundary Conditions ◽

Well Posedness ◽

Time Behavior ◽

Long Time Behavior ◽

Dynamic Boundary Conditions ◽

Finite Dimensional ◽

Dynamic Boundary ◽

Long Time ◽

Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

Download Full-text

Inverse problem for a degenerate/singular parabolic system with Neumann boundary conditions

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2018-0034 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mohammed Alaoui ◽

Abdelkarim Hajjaj ◽

Lahcen Maniar ◽

Jawad Salhi

Keyword(s):

Boundary Conditions ◽

Parabolic System ◽

Source Term ◽

Neumann Boundary ◽

Stability Estimate ◽

Carleman Estimates ◽

Well Posedness ◽

Degenerate Diffusion ◽

Neumann Type ◽

Lower Order Terms

AbstractIn this paper, we study an inverse source problem for a degenerate and singular parabolic system where the boundary conditions are of Neumann type. We consider a problem with degenerate diffusion coefficients and singular lower-order terms, both vanishing at an interior point of the space domain. In particular, we address the question of well-posedness of the problem, and then we prove a stability estimate of Lipschitz type in determining the source term by data of only one component. Our method is based on Carleman estimates, cut-off procedures and a reflection technique.

Download Full-text