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Abdelhafid Younsi

Publications and source records attributed to Abdelhafid Younsi.

7 recordsLinked to original sources
arXiv2016

Lower bound on the blow-up rate of the 3D Navier-Stokes equations in H^{5/2}

Under assumption that $T^{\ast}$ is the maximal time of existence of smooth solution of the 3D Navier-Stokes equations in the Sobolev space $H^{s}$, we establish lower bounds for the blow-up rate of the type$\ \left( T^{\ast }-t\right) ^{-φ\left( n\right) }$, where $n$ is a natural number independent of $s$ and $φ$ is a linear function. Using this new type in the 3D Navier-Stokes equations in the $H^{5/2}$, both on the whole space and in the periodic case, we give an answer to a question left open by James et al (2012, J. Math. Phys.). We also prove optimal lower bounds for the blow-up rate in $\dot{H}^{3/2}$ and in $H^{1}$.

math.AP↗
arXiv2016

Uniqueness criterion of weak solutions for the 3D Navier-Stokes equations

In this paper we establish a new uniqueness result of weak solutions for the 3D Navier-Stokes equations. Under assumption that there is not uniqueness of weak solution in singular time, we prove that if two weak solutions $u$ and $v$ of 3D Navier-Stokes equations belong to $L^{\frac{5}{2}}\left( 0,T;V\right) $ with the same initial datum, then we get $u=v$. In the class $L^{3}\left( 0,T;V\right) $, we prove that $u-v\in$ $C^{0}\left( 0,T;H\right) $ and $u=v\ $when $u_{0}=v_{0}$.

math.AP↗
arXiv2012

Upper semicontinuous attractors for 3D hyperviscous flow

We regularized the 3D Navier-Stokes equations by adding a fourth-order viscosity term. We study the upper semicontinuity, of the global attractors of the Leray-Hopf weak solutions of the regularized 3D Navier-Stokes equations, as the artificial dissipation goes to 0. We also consider applications of obtained results to the regularized problem by allowing the family of forcing functions to vary with epsilon, for epsilon >0.

math.AP↗
arXiv2010

Hausdorff measure of Vorticity Nodal Sets for the 3D Hyperviscous Navier Stokes Equations with General forces

In this paper, we modified the three dimensional Navier-Stokes equations by adding a l-Laplacian. We provide upper bounds on the two-dimensional Hausdorff measure the level sets of the vorticity of solutions. We express them in terms of the Kolmogorov length-scale and the Landau-Lifschitz estimates of the number of degrees of freedom in turbulent flow. We also, under certain hypothesis recover the two-dimensional Hausdorff measure estimates for the usual 3D Navier-Stokes equations with potential force. Moreover, we show that the estimates depend on l, this result suggests that the modified Navier Stokes system is successful model of turbulence and the size of the nodal set leads the way for developing the turbulence theory.

math.AP↗
arXiv2010

Effect of hyperviscosity on the Navier-Stokes turbulence

In this paper we modified the Navier-Stokes equations by adding a higher order artificial viscosity term to the conventional system. We first show that the solution of the regularized system converges strongly to the solution of the conventional system as the regularization parameter goes to zero, for each dimension d $\le$ 4. Then we show that the use of this artificial viscosity term leads to truncated the number of degrees of freedom in the long-time behavior of the solutions to these equations. This result suggests that the hyperviscous Navier-Stokes system is an interesting model for three-dimensional fluid turbulence.

math.AP↗