On the regularity for solutions of the micropolar fluid equations
On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations.
We derive various estimates for strong solutions to the Navier-Stokes equations in C([0,T),L3(R3)) that allow us to prove some regularity results on the kinematic bilinear term.
On the result of He concerning the smoothness of solutions to the Navier-Stokes equations.
On the search for singularities in incompressible flows
In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.
On the second order slip Reynolds equation with molecular dynamics: existence and uniqueness.
On the smallness of the (possible) singular set in space for 3D Navier-Stokes equations.
On the structure of flows through pipe-like domains satisfying a geometrical constraint
We study solutions of the steady Navier-Stokes equations in a bounded 2D domain with the slip boundary conditions admitting flow across the boundary. We show conditions guaranteeing uniqueness of the solution. Next, we examine the structure of the solution considering an approximation given by a natural linearization. Suitable error estimates are also obtained.
On various types of viscous two-fluid flows
Viscous two-fluid flows arise in different kinds of coating technologies. Frequently, the corresponding mathematical models represent two-dimensional free boundary value problems for the Navier-Stokes equations or their modifications. In this review article we present some results about nonisothermal stationary as well as about isothermal evolutionary viscous flow problems. The temperature-depending problems are characterized by coupled heat- and mass transfer and also by thermocapillary convection....
On weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids: defect measure and energy equality
We consider the non-stationary Navier-Stokes equations completed by the equation of conservation of internal energy. The viscosity of the fluid is assumed to depend on the temperature, and the dissipation term is the only heat source in the conservation of internal energy. For the system of PDE's under consideration, we prove the existence of a weak solution such that: 1) the weak form of the conservation of internal energy involves a defect measure, and 2) the equality for the total energy is satisfied....