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The formation of singularities in finite time in non-local Burgers equations, with time-fractional derivative, is studied in detail. The occurrence of finite time singularity is proved, revealing the underlying mechanism, and precise estimates on the blow-up time are provided. The employment of the present equation to model a problem arising in job market is also analyzed.
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This paper is concerned with the localized behaviors of the solution $u$ to the Navier-Stokes equations near the potential singular points. We establish the concentration rate for the $L^{p,infty}$ norm of $u$ with $3leq pleqinfty$. Namely, we show that if $z_0=(t_0,x_0)$ is a singular point, then for any $r>0$, it holds begin{align} limsup_{tto t_0^-}||u(t,x)-u(t)_{x_0,r}||_{L^{3,infty}(B_r(x_0))}>delta^*, otag end{align} and begin{align} limsup_{tto t_0^-}(t_0-t)^{frac{1}{mu}}r^{frac{2}{ u}-frac{3}{p}}||u(t)||_{L^{p,infty}(B_r(x_0))}>delta^* otag for~3<pleqinfty, ~frac{1}{mu}+frac{1}{ u}=frac{1}{2}~and~2leq uleqfrac{2}{3}p, otag end{align}where $delta^*$ is a positive constant independent of $p$ and $ u$. Our main tools are some $varepsilon$-regularity criteria in $L^{p,infty}$ spaces and an embedding theorem from $L^{p,infty}$ space into a Morrey type space. These are of independent interests.
We study the problem of global exponential stabilization of original Burgers equations and the Burgers equation with nonlocal nonlinearities by controllers depending on finitely many parameters. It is shown that solutions of the controlled equations are steering a concrete solution of the non-controlled system as $trightarrow infty$ with an exponential rate.
In this paper, we present strong numerical evidences that the $3$D incompressible axisymmetric Navier-Stokes equations with degenerate diffusion coefficients and smooth initial data of finite energy develop a potential finite time locally self-similar singularity at the origin. The spatial part of the degenerate diffusion coefficient is a smooth function of $r$ and $z$ independent of the solution and vanishes like $O(r^2)+O(z^2)$ near the origin. This potential singularity is induced by a potential singularity of the $3$D Euler equations. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the property that the center of the traveling wave approaches the origin at a slower rate than the rate of the collapse of the singularity. The driving mechanism for this potential singularity is due to two antisymmetric vortex dipoles that generate a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the $3$D Euler equations develop an additional small scale characterizing the thickness of the sharp front. On the other hand, the Navier-Stokes equations with a constant diffusion coefficient regularize the two-scale solution structure and do not develop a finite time singularity for the same initial data. The initial condition is designed in such a way that it generates a positive feedback loop that enforces a strong nonlinear alignment of vortex stretching, leading to a stable locally self-similar blowup at the origin. We perform careful resolution study and asymptotic scaling analysis to provide further support of the potential finite time locally self-similar blowup.
Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blowup scenario revealed by Luo-Hou in cite{luo2014potentially,luo2014toward}, which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou-Huang in cite{Hou-Huang-2021}. One important difference between these two blowup scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in cite{Hou-Huang-2021}. More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier--Stokes equations. We will present strong numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data. Finally, we present some preliminary results to demonstrate that the 3D Navier--Stokes equations using the same initial condition develop nearly singular behavior with maximum vorticity increased by a factor of $10^{7}$.
We investigate the soliton dynamics for the fractional nonlinear Schrodinger equation by a suitable modulational inequality. In the semiclassical limit, the solution concentrates along a trajectory determined by a Newtonian equation depending of the fractional diffusion parameter.
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