Subcritical Spaces Research Articles

Well-posedness and stability for the two-phase periodic quasistationary Stokes flow

The two-phase horizontally periodic quasistationary Stokes flow in \mathbb{R}^{2} , describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, parameterized as the graph of a function f=f(t) , is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function f . Based on abstract parabolic theory, it is shown that the problem is well-posed in all subcritical spaces \mathrm{H}^{r}(\mathbb{S}) , with r\in(3/2,2) . Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.

Borderline gradient continuity for fractional heat type operators

In this paper, we establish gradient continuity for solutions to \[ (\partial_t - \operatorname{div}(A(x) \nabla ))^{s} u =f,\quad s \in (1/2, 1), \]when $f$ belongs to the scaling critical function space $L\left (\frac {n+2}{2s-1}, 1\right )$. Our main results theorems 1.1 and 1.2 can be seen as a nonlocal generalization of a well-known result of Stein in the context of fractional heat type operators and sharpen some of the previous gradient continuity results which deal with $f$ in subcritical spaces. Our proof is based on an appropriate adaptation of compactness arguments, which has its roots in a fundamental work of Caffarelli in [13].

Well-posedness of the Muskat problem in subcritical Lp-Sobolev spaces

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

Random initial conditions for semi-linear PDEs

AbstractWe analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

Local well‐posedness of critical nonlinear Schrödinger equation on Zoll manifolds of odd‐growth

In this paper, we study the nonlinear Schrödinger equation on Zoll manifolds with odd order nonlinearities. We will obtain the local well‐poesdness in the critical space . This extends the recent results in the literature to the Zoll manifolds of dimension d≥2 with general odd order nonlinearities and also partially improves the previous results in the subcritical spaces of Yang to the critical cases. Copyright © 2015 John Wiley & Sons, Ltd.

Subcritical space nuclear system without most movable control systems

This paper describes the design and analysis of advanced space nuclear reactor (ASNR) whose design combines the advantages of radioisotope thermoelectric generator (RTG) and space nuclear reactor (SNR). As opposed to current SNRs designs, ASNR is a subcritical system driven by 232U–Be neutron source to generate thermal power continuously. Most movable control systems in the SNR design are removed. The detailed neutronic calculations by MCNPX (Monte Carlo N-Particle eXtended), including keff, flux, burn-up, loss-ratio of neutron source and immersion reactivity, show that ASNR has higher criticality safety and more compact structure to bear the risk of immersion accident compared with the past SNRs, and the new system can provide more thermal power than RTG. Furthermore, the neutron source efficiency is optimized to improve the utilization of 232U–Be neutron source with the improvement of criticality safety. Compared with the past designs of space nuclear power, ASNR could provide enough thermal power and avoid the occurrence of serious immersion accident in the case of total control system failure. ASNR has potential for future deep space missions.

Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

Abstract The well-posedness for the Cauchy problem of the nonlinear fractional Schrödinger equation $u_t + i( - \Delta )^\alpha u + i|u|^2 u = 0,(x,t) \in \mathbb{R}^n \times \mathbb{R},\frac{1} {2} < \alpha < 1 $ is considered. The local well-posedness in subcritical space H s with s > n/2 -α is obtained. Moreover, the inviscid limit behavior of solution for the fractional Ginzburg-Landau equation $u_t + (\nu + i)( - \Delta )^\alpha u + i|u|^2 u = 0$ is also considered. It is shown that the solution of the fractional Ginzburg-Landau equation converges to the solution of nonlinear fractional Schrödinger equation in the natural space C([0, T];H)s) with s > n/2 — α if ν tends to zero.

On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations

In this paper, we intend to reveal how the fractional dissipation (−Δ) α affects the regularity of weak solutions to the 3d generalized Navier- Stokes equations. Precisely, it will be shown that the (5 − 4α)/2α dimensional Hausdorff measure of possible time singular points of weak solutions on the interval (0, ∞ )i s zero when 5/6 ≤ α< 5/4. To this end, the eventual regularity for the weak solutions is firstly established in the same range of α .I t is worth noting that when the dissipation index α varies from 5/ 6t o 5/4, the corresponding Hausdorff dimension is from 1 to 0. Hence, it seems that the Hausdorff dimension obtained is optimal. Our results rely on the fact that the space H α is the critical space or subcritical space to this system when α ≥ 5/6.