Related papers: Some interesting class of integrable partial diffe…
We analyze the forced incompressible stationary Navier-Stokes flow in $\mathbb{R}^n_+$, $n>2$. Existence of a unique solution satisfying a global integrabilty property measured in a scale of tent spaces is established for small data in…
This note introduces a novel numerical analysis framework for the incompressible Navier-Stokes equations based on Besov spaces. The key contribution of this note is to establish the stability and convergence of a semi-implicit time-stepping…
In this study, the existence and uniqueness of the unpredictable solution for a non-homogeneous linear system of ordinary differential equations is considered. The hyperbolic case is under discussion. New properties of unpredictable…
Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature are hard to find computationally. We present a family of methods to converge such unstable periodic orbits for the incompressible…
Using exhaustion method and finite differences a new method to solve system of partial differential equations and is presented. This method allows design algorithm to solve linear and nonlinear systems in irregular domains. Applying this…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
Symmetry reductions of systems of two nonlinear partial differential equations are studied. We find ansatzes reducing system of partial differential equations to system of ordinary differential equations. The method is applied to system…
Whether the 3D incompressible Navier-Stokes equations will have a global smooth solution for all smooth, finite energy initial data is a Millennium Prize problem. One of the main difficulties of this problem is that the Navier-Stokes…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…
A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting…
Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years,…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
The paper concerns with the global well-posedness issue of the 2D incompressible inhomogeneous Navier-Stokes (INS) equations with fractional dissipation and rough density. We first establish the $L^q_t(L^p)$-maximal regularity estimate for…
We first study the linear eigenvalue problem for a planar Dirac system in the open half-line and describe the nodal properties of its solution by means of the rotation number. We then give a global bifurcation result for a planar nonlinear…
We show existence and regularity result for the Navier Stokes system for small data in the space $\Phi(2)$, and we show relations with some classical results.
In this paper we use the generalized tanh method to obtain exact solutions for a class of fifth-order nonlinear systems. A particular case is give by the integrable Mikhailov--Novikov--Wang system (MNW). Periodic and soliton solutions are…
In this paper we propose new method for proving of global solutions for 3D Navier-Stokes equations. This complies an application to the Clay Institute Millennium Prize Navier Stokes Problem. The proposed method can be applied for…
We investigate existence of solitonic solutions for higher-order partial differential equations with polynomial nonlinearities. Using the Hirota method we obtain classification for higher-order integrable systems of equations.