Second order fractional mean-field SDEs with singular kernels and measure initial data

Abstract

In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton or Coulomb potential, Riesz potential, Biot-Savart law, etc. Moreover, we also show the stability, smoothness and the short time singularity and large time decay estimates of the distribution density. Our results reveal a phenomenon that for nonlinear mean-field equations, the regularity of the initial distribution could balance the singularity of the kernel. The precise relationship between the singularity of kernels and the regularity of initial value are calculated, which belongs to the subcritical regime in scaling sense. In particular, our results provide microscopic probability explanation and establish a unified treatment for many physical models such as fractional Vlasov-Poisson-Fokker-Planck system, the vorticity formulation of 2D-fractal Navier-Stokes equations, surface quasi-geostrophic models, fractional porous medium equation with viscosity, etc.

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

“…Additive noise was invoked as the most natural candidate for the regularization of noise. One main reason is that it succeeds in improving well-posedness of finite-dimensional stochastic differential equations with irregular drifts [21,35,48,58]. This phenomena also exhibit for certain infinite-dimensional SPDEs [7,[18][19][20].…”

“…Equations of fluid mechanics driven by stochastic force have been studied extensively in literature. Since the work [6] by Bensoussan-Temam in the early 70's, many aspects of stochastic NSE have been developed, including the existence of martingale solutions [8,12,28,56], ergodicity [34,47] and stochastic representation [15,35].…”

Propagation of chaos and conditional McKean-Vlasov SDEs with regime-switching

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We study the propagation of chaos in a class of moderately interacting particle systems for the approximation of singular kinetic McKean-Vlasov SDEs driven by α-stable processes. Diffusion parts include Brownian (α = 2) and pure-jump (α ∈ (1, 2)) perturbations and interaction kernels are considered in a non-smooth anisotropic Besov space.Using Duhamel formula, sharp density estimates (recently issued in [17]), and suitable martingale functional inequalities, we obtain direct estimates on the convergence rate between the empirical measure of the particle systems toward the McKean-Vlasov distribution. These estimates further lead to quantitative propagation of chaos results in the weak and strong sense.

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“…Weak and strong wellposedness of (1.1) have been established in [17], in the case where t → b t lies in a (negatively smooth) mixed Besov space:…”

“…intrinsic scales of the underlying kinetic system). Notably, the authors in [17] established that µ t sits in a balanced duality with b, the resulting drift component b * µ belonging to L s ([0, T ]; L ∞ (R 2d )) for some appropriate s > 2. (See also [8,9] for similar results in the non-degenerate setting.…”