Abstract In the context of integrable difference equations on quad-graphs, we introduce method open boundary reductions, as an alternative to well-known periodic for constructing discrete mappings and their invariants. The are obtained from well-posed initial value problems quad restricted strips ${{\mathbb{Z}}}^2$-lattices. invariants constructed using Sklyanin’s double-row monodromy matrix. T...

In this paper we study finite dimensional approximations to Boussinesq type equations. Our methods are based on infinite dimensional center manifold theory. The main advantage of our approach is that we can handle both well-posed and ill-posed versions of the Boussinesq equation. We show that for suitable initial conditions, our approximations describe the dynamics accurately for long enough ti...

In a recent paper [12], Vishik proved the global wellposedness of the two-dimensional Euler equation in the critical Besov space B 2,1. In the present paper we prove that the Navier-Stokes system is globally well-posed in B 2,1, with uniform estimates on the viscosity. We prove also a global result of inviscid limit. The convergence rate in L is of order ν.

In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.

In this paper we study a fractional diffusion Boussinesq model which couples a Navier-Stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature. We establish global well-posedness results with rough initial data.

This paper studies the global well-posedness of the incompressible magnetohydrodynamic (MHD) system with a velocity damping term. We establish the global existence and uniqueness of smooth solutions when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also given.

We study the well-posedness and regularity of the generalized Navier-Stokes equations with initial data in a new critical space Q α;∞ (R ) = ∇ · (Qα(R )), β ∈ ( 1 2 , 1) which is larger than some known critical homogeneous Besov spaces. Here Qα(R ) is a space defined as the set of all measurable functions with sup(l(I)) Z

In this paper we obtain new well-possedness results concerning a linear inhomogenous Stokes-like system. These results are used to establish local well-posedness in the critical spaces for initial density ρ0 and velocity u0 such that ρ0−ρ ∈ Ḃ 3 p p,1(R ), u0 ∈ Ḃ 3 p −1 p,1 (R ), p ∈ ( 6 5 , 4 ) , for the inhomogeneous incompressible Navier-Stokes system with variable viscosity. To the best of o...

We prove that the Cauchy problem for the three-dimensional Navier–Stokes equations is ill-posed in Ḃ −1,∞ ∞ in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small in Ḃ−1,∞ ∞ can produce solutions arbitrarily large in Ḃ−1,∞ ∞ after an arbitrarily short time. Such a result implies that the solution ...

In this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations. With those, well-posedness can be proven for a general initial-boundary value problem. The procedure is exemplified on the linearized Navier–Stokes equations in two and three space dimensions on a general domain.