687 research outputs found
Boussinesq Equations with Partial or Fractional Dissipation
The two-dimensional (2D) incompressible Boussinesq system is not only an important model in geophysics, but also retains some key features of the 3D Euler and Navier-Stokes equations such as the vortex stretching mechanism. Especially, the inviscid 2D Boussinesq equations are identical to the Euler equations for the 3D axisymmetric swirling flows. Even though the global regularity of full dissipative Boussinesq equations is well known, the global regularity problem of inviscid case is still left open. First, we prove the global existence and uniqueness of 2D Boussinesq equations with partial dissipation in bounded main with Navier type boundary conditions. Secondly, we investigate Boussinesq equations with fractional dissipation on a d-dimensional periodic domain, and apply a re-developed tool of LittlewoodPaley decomposition to achieve global existence and uniqueness of weak solutions. Lastly, we focus on several variants of the 2D incompressible Euler equations. It is not known whether global well-posedness result would hold if there is only partially damping term for 2D Euler equation. Besides, in the vorticity equations, the partially damping term becomes a non-local operator \mathcal R_2^2 \omega. Our numerical simulations show that by replacing \mathcal R_2^2 \omega with different operators (e.g. \mathcal R_1\mathcal R_2 \omega), the solutions will behave quite differently
Existence and uniqueness of global solutions for the modified anisotropic 3D Navier-Stokes equations
We study a modified three-dimensional incompressible anisotropic
Navier-Stokes equations. The modification consists in the addition of a power
term to the nonlinear convective one. This modification appears naturally in
porous media when a fluid obeys the Darcy-Forchheimer law instead of the
classical Darcy law. We prove global in time existence and uniqueness of
solutions without assuming the smallness condition on the initial data. This
improves the result obtained for the classical 3D incompressible anisotropic
Navier-Stokes equations.Comment: To appear in ESAIM: Mathematical Modelling and Numerical Analysi
On a Navier-Stokes-Allen-Cahn model with inertial effects
A mathematical model describing the flow of two-phase fluids in a bounded
container is considered under the assumption that the phase transition
process is influenced by inertial effects. The model couples a variant of the
Navier-Stokes system for the velocity with an Allen-Cahn-type equation for
the order parameter relaxed in time in order to introduce inertia.
The resulting model is characterized by second-order material derivatives which
constitute the main difficulty in the mathematical analysis. Actually, in order
to obtain a tractable problem, a viscous relaxation term is included in the
phase equation. The mathematical results consist in existence of weak solutions
in 3D and, under additional assumptions, existence and uniqueness of strong
solutions in 2D. A partial characterization of the long-time behavior of
solutions is also given and in particular some issues related to dissipation of
energy are discussed.Comment: 24 page
Mathematical Aspects of Hydrodynamics
The workshop dealt with the partial differential equations that describe fluid motion and related topics.
These topics included both inviscid and viscous fluids in two and three dimensions. Some talks addressed
aspects of fluid dynamics such as the construction of wild weak solutions, compressible shock formation,
inviscid limit and behavior of boundary layers, as well as both polymer/fluid and structure/fluid interaction
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