687 research outputs found

    Boussinesq Equations with Partial or Fractional Dissipation

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    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

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    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

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    A mathematical model describing the flow of two-phase fluids in a bounded container Ω\Omega 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 uu with an Allen-Cahn-type equation for the order parameter φ\varphi 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

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    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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