This paper is concerned with the pullback dynamics and robustness for the 2D incompressible Navier-Stokes equations with delay on the convective term in bounded domain. Under appropriate assumption on the delay term, we establish the existence of pullback attractors for the fluid flow model, which is dependent on the past state. Inspired by the idea in Zelati and Gal’s paper (JMFM, 2015), the robustness of pullback attractors has been proved via upper semi-continuity in last section.
In this work, we study the Cauchy problem of the spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework. We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L2 initial datum for positive time. So that the smoothing effect of the Cauchy problem for the spatially homogeneous Landau equation with hard potentials is exactly same as heat equation.
We prove the existence of multiple solutions of an elliptic equation with critical Sobolev growth and critical Hardy potential on compact Riemannian manifolds.
The viability of the conformable stochastic differential equations is studied. Some necessary and sufficient conditions in terms of the distance function to K are given. In addition, when the boundary of K is sufficiently smooth, our necessary and sufficient conditions can reduce to two relations just on the boundary of K. Lastly, an example is given to illustrate our main results.
The biwave maps are a class of fourth order hyperbolic equations. In this paper, we are interested in the solution formulas of the linear homogeneous biwave equations. Based on the solution formulas and the weighted energy estimate, we can obtain the ${{L}^{\infty }}({{\mathbb{R}}^{n}})-{{W}^{N,1}}({{\mathbb{R}}^{n}})$ and ${{L}^{\infty }}({{\mathbb{R}}^{n}})-{{W}^{N,2}}({{\mathbb{R}}^{n}})$ estimates, respectively. By our results, we find that the biwave maps enjoy some different properties compared with the standard wave equations.
In this paper, we study the quenching phenomenon of solutions for parabolic equations with singular absorption under the mixed boundary conditions on graphs. Firstly, we prove the local existence of solutions via Schauder fixed point theorem. Then, under certain conditions we obtain the estimates of quenching time and quenching rate. Finally, numerical experiments are shown to explain the theoretical results.