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We consider some reaction-diffusion equations describing systems with the nonlocal consumption of resources and the intraspecific competition. Sharp conditions on the coefficients are obtained to ensure the stability and instability of…
This article reviews the properties of the self-similar solutions of the Navier-Stokes equation for incompressible fluids. Since any smooth solution can be embedded into a self-similar solution at the identity scale, it follows that under…
In this paper we study the existence of solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids.…
We study the existence of critical points of stable stationary solutions to reaction-diffusion problems on topological tori. Stable nonconstant stationary solutions are often called patterns. We construct topological tori and patterns with…
We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set…
The present paper studies the structure of the set of stationary solutions to the incompressible Euler equations on the rotating unit sphere that are near two basic zonal flows: the zonal Rossby-Haurwitz solution of degree 2 and the zonal…
Rotating and twisting locally rotationally symmetric imperfect fluids in general relativity admit a much larger set of solutions than the self-similar ones recently suggested in the literature. Explicit forms of the metrics are given and…
Stability of self-similar solutions for gravitational collapse is an important problem to be investigated from the perspectives of their nature as an attractor, critical phenomena and instability of a naked singularity. In this paper we…
New 2-norm bounds for solutions of planar div-curl boundary value problems on bounded planar regions are described. Prescribed flux, tangential trace and mixed boundary boundary are treated. A harmonic decomposition is used to separate…
Ideal fluid dynamics is studied as a relativistic field theory with particular importance on its hamiltonian structure. The Schwinger condition, whose integrated version yields the stress tensor conservation, is explicitly verified both in…
The collapsing dynamics of relativistic fluid are explored in $f(R)$ gravity in a detailed systematic manner for the non-static spherically symmetric spacetime satisfying the equation of the conformal Killing vector. With quasi-homologous…
Stationary rotating matter configurations in general relativity are considered. A formalism for general stationary space times is developed. Axisymmetric systems are discussed by the use of a nonholonomic and nonrigid frame in the…
Spherical accretion flows are simple enough for analytical study, by solution of the corresponding fluid dynamic equations. The solutions of stationary spherical flow are due to Bondi. The questions of the choice of a physical solution and…
The motion of compressible, inviscid fluid under the constant pressure on a rotating sphere is studied. The hodograph equations for the corresponding Euler equation are presented. They provide us with the class of solutions of the Euler…
Exact solutions of a classical problem of a plane unsteady potential flow of an ideal incompressible fluid with a free boundary are presented. The fluid occupies a semi-infinite strip bounded by the free surface (from above) and (from the…
We study different dimensional fluids inspired by noncommutative geometry which admit conformal Killing vectors. The solutions of the Einstein field equations examined specifically for five different set of spacetime. We calculate the…
By folding nonautonomous differential systems in the plane to scalar differential equations, a sufficient condition for the non-occurrence of chaotic behavior is obtained.
A class of stationary rigidly rotating perfect fluid coupled with non-linear electromagnetic fields was investigated. An exact solution of the Einstein equations with sources for the Carter B(+) branch was found, for the equation of state…
It is proven that, under mild physical assumptions, an isolated stationary relativistic perfect fluid consists of a finite number of cells fibred by invariant annuli or invariant tori. For axially symmetric circular flows it is shown that…
A new approach is described to help improve the foundations of relativistic viscous fluid dynamics and its coupling to general relativity. Focusing on neutral conformal fluids constructed solely in terms of hydrodynamic variables, we derive…