Related papers: Almost sure existence of global solutions for gene…
We prove global existence and blow-up of solutions of initial-boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for…
In this paper, complex Ginzburg-Landau (CGL) equations governed by p-Laplacian are studied. We discuss the global existence of solutions for the initial-boundary value problem of the equation in general domains. The global solvability of…
We investigate the global-in-time existence and uniqueness of weak solutions for a family of equations introduced by Moffatt to model magnetic relaxation. These equations are topology-preserving and admit all stationary solutions to the…
In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a…
The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…
Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition.
Considering the three-dimensional incompressible Navier-Stokes equations on the whole space, we address the question: is it possible to infer global regularity of a mild solution from a single approximate solution? Assuming a relatively…
We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on $(R^3, g)$, where the metric $g$ is a small perturbation of the flat metric and approaches the…
Consider the Cauchy problem of incompressible Navier-Stokes equations in $\mathbb{R}^3$ with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the…
In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $\dot{H}^{-\alpha}(\mathbb{R}^{3})$ or $\dot{H}^{-\alpha}(\mathbb{T}^{3})$ with…
In this paper, we establish the global existence of smooth solutions to general 4D quasilinear wave equations satisfying the first null condition with the short pulse initial data. Although the global existence of small data solutions to 4D…
In this work we consider the initial-value problem associated with a coupled system of generalized Korteweg-de Vries equations. We present a relationship between the best constant for a Gagliardo-Nirenberg type inequality and a criterion…
Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad &…
In this paper we introduce a probabilistic approach to show the existence of initial data with arbitrarily large $L^2(\mathbb{R}^3)$, $\dot{H}^{1/2}(\mathbb{R}^3)$ and $\mathcal{PM}^2$-norms for which a Generalized Navier-Stokes system…
In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier-Stokes equations with initial data in critical Besov spaces, which satisfies a non-linear smallness condition. The…
We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schr\"odinger (NLS) and nonlinear wave (NLW) equations on the unit ball in R^d to the case…
We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $\gamma\ge1$. Initial data and solutions are small in $L^2$ around a…
Initial boundary value problems for the generalized Benney-Lin equation posed on bounded intervals and on the right half-line were considered. The existence and uniqueness of global regular solutions on arbitrary intervals as well as their…
We find a global a priori estimate for solutions to the Navier-Stokes equations with periodic boundary conditions guaranteeing in view of the Serrin type condition the existence of global regular solutions. We derive the following estimate…
We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in $\H{m}$, for $m$ large enough. To do so, we introduce a specific symmetrisation allowing areas of…