Consider the Cauchy problem for the $n$-dimensional incompressible Navier-Stokes equations. There have been very interesting advances on the mathematical influences of physical mechanisms on the global smooth solutions (if the dimension $n=2$) and the global weak solutions (if the dimension $n\geq 3$) of the Navier-Stokes equations. This paper is a simple survey of recent advances on the properties of the solutions.
In this paper, we investigate the long term dynamics of the McKean-Vlasov stochastic $p$-Laplace lattice system with time delay driven by Lévy noise in a product space instead of the Skorohod space. We first define a non-autonomous dynamical system in the space of probability measures in the product space and prove its continuity over bounded sets for all time. This is in sharp contrast with the dynamical system defined in the Skorohod space which is continuous only for large time. We then establish the existence and uniqueness of measure attractors in the product space and prove the upper semi-continuity of these attractors when the noise intensity approaches zero.
We study a symmetric class of three-dimensional compressible current-vortex sheets for which the basic velocity is tangential to the interface and the basic magnetic field is tangential to the interface but transverse to the velocity. Restricting to perturbations that are independent of the magnetic-field direction, we derive an exact scalar pseudo-differential equation for the interface. If the magneto-acoustic Mach number $$M_B:=\frac{|\dot{v}_1^+|}{\bar{C}_B},\quad \bar{C}_B^2=c^2+\frac{|\dot{B}_2|^2}{\dot{\varrho}}$$, satisfies $M_B>\sqrt{2}$, we prove unique solvability of the reduced interface equation and the weighted estimate $$\gamma^3\|f\|_{H_\gamma^{s+1}}^2\leq C\sum\limits_{\pm}\|\mathcal{K}^{\pm}\|_{L^2(\mathbb{R}_{\pm};H_\gamma^s)}^2,\quad\gamma\geq1$$. This result complements the low-$M_B$ linear and nonlinear ill-posedness established by Xie and Zhao in the Journal of Geometric Analysis [32].
We prove nonuniqueness of distributional weak solutions for the flux-limited diffusion equation. For compactly supported positive data, the entropy condition selects saturated discontinuity fronts moving with light speed. If, instead, one prescribes strictly sub-saturated boundary fluxes in the mass-coordinate dual problem, the dual equation is locally uniformly parabolic and has a classical solution for short time. The inverse mass transformation gives a piecewise classical weak solution whose free boundaries satisfy the Rankine–Hugoniot condition but move with the prescribed sub-light speeds. This solution has the same initial datum as the entropy solution and differs from it immediately after the initial time. In particular, the entropy solution is also a weak solution; hence weak solutions are nonunique.
