The Dirchlet Problem for a Complex Monge-Ampère Equation.
The effect of a magnetic field on the onset of Bénard convection in variable viscosity couple-stress fluids using classical Lorenz model
The Rayleigh-Bénard convection for a couple-stress fluid with a thermorheological effect in the presence of an applied magnetic field is studied using both linear and non-linear stability analysis. This problem discusses the three important mechanisms that control the onset of convection; namely, suspended particles, an applied magnetic field, and variable viscosity. It is found that the thermorheological parameter, the couple-stress parameter, and the Chandrasekhar number influence the onset of...
The Effect of Bacteria on Epidermal Wound Healing
Epidermal wound healing is a complex process that repairs injured tissue. The complexity of this process increases when bacteria are present in a wound; the bacteria interaction determines whether infection sets in. Because of underlying physiological problems infected wounds do not follow the normal healing pattern. In this paper we present a mathematical model of the healing of both infected and uninfected wounds. At the core of our model is an...
The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations.
The effect of thin coatings on second harmonic generation.
The effective boundary conditions for vector fields on domains with rough boundaries: Applications to fluid mechanics
The Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets. Assuming the complete slip boundary conditions we identify the limit system, in particular, we show that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblets.
The eigenvalue problem and Jacobean-like methods for its solution
The elliptic problems in a family of planar open sets
We propose, on a model case, a new approach to classical results obtained by V. A. Kondrat'ev (1967), P. Grisvard (1972), (1985), H. Blum and R. Rannacher (1980), V. G. Maz'ya (1980), (1984), (1992), S. Nicaise (1994a), (1994b), (1994c), M. Dauge (1988), (1990), (1993a), (1993b), A. Tami (2016), and others, describing the singularities of solutions of an elliptic problem on a polygonal domain of the plane that may appear near a corner. It provides a more precise description of how the solutions...
The energy of heavy atoms according to Brown and Ravenhall: the Scott correction.
The equations of viscous incompressible nonhomogeneous fluids in noncylindrical domains: on the existence and regularity
We prove the existence of a weak solution and of a strong solution (locally in time) of the equations which govern the motion of viscous incompressible non-homogeneous fluids. Then we discuss the decay problem.
The essential spectrum of relativistic multi-particle operators
The essential spectrum of relativistic one-electron ions in the Jansen-Hess model.
The Eulerian limit and the slip boundary conditions-admissible irregularity of the boundary
We investigate the inviscid limit for the stationary Navier-Stokes equations in a two dimensional bounded domain with slip boundary conditions admitting nontrivial inflow across the boundary. We analyze admissible regularity of the boundary necessary to obtain convergence to a solution of the Euler system. The main result says that the boundary of the domain must be at least C²-piecewise smooth with possible interior angles between regular components less than π.
The evolution Darcy-Boussinesq system (a weak maximum principle and the uniqueness)
The evolution of dust emitted by a uniform source above ground level.
The existence and uniqueness of solutions of equations for ideal compressible polytropic fluids.
The existence of a solution and a numerical method for the Timoshenko nonlinear wave system
The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version of...
The existence of a solution and a numerical method for the Timoshenko nonlinear wave system
The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version...
The existence of an exponential attractor in magneto-micropolar fluid flow via the ℓ-trajectories method
We consider the magneto-micropolar fluid flow in a bounded domain Ω ⊂ ℝ². The flow is modelled by a system of PDEs, a generalisation of the two-dimensional Navier-Stokes equations. Using the Galerkin method we prove the existence and uniqueness of weak solutions and then using the ℓ-trajectories method we prove the existence of the exponential attractor in the dynamical system associated with the model.
The existence of Ginzburg-Landau solutions on the plane by a direct variational method