Journal de Mathematiques Pures et Appliquees最新文献

We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys $C^{1,1}$ regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be Hölder continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

We prove a Cwikel–Lieb–Rozenblum type inequality for the number of negative eigenvalues of the Hardy–Schrödinger operator $-\Delta -{(d-2)}^2/\left(4\right|x{|}^2)-W(x)$ on $L^2\left(R^d\right)$. The bound is given in terms of a weighted $L^{d/2}$-norm of W which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.

We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot–Chow morphism of tensor and wedge products of duals of tautological bundles.

This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [47], Lim–Zhou [38], and Yong [72]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman–Kac formula for BSVIEs and some stability estimates of the representation parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein–Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [47], [38], [72], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [73], [75] and Björk–Khapko–Murgoci [6].

We establish a Talenti-type symmetrization result in the form of mass concentration (i.e. integral comparison) for very general linear nonlocal elliptic problems, equipped with homogeneous Dirichlet boundary conditions.

In this framework, the relevant concentration comparison for the classical fractional Laplacian can be reviewed as a special case of our main result, thus generalizing the previous results in [20].

Finally, using an implicit time discretization techniques, similar results are obtained for the solutions of Cauchy-Dirichlet nonlocal linear parabolic problems.

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any $L^2$ divergence-free initial data, there exist unique smooth Leray-Hopf solutions when $\alpha \ge 5/4$. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces $L_t^{\gamma }{W}_x^{s,p}$, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints $(3/p+1-2\alpha ,\infty ,p)$ and $(2\alpha /\gamma +1-2\alpha ,\gamma ,\infty )$. Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff $H^{{\eta }_{⁎}}$ measure, where ${\eta }_{⁎}>0$ is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.

In the setting of horizontal curves in the Heisenberg group, we prove a $C^{m,\omega }$ finiteness principle, a $C^{m,\omega }$ Lusin approximation result, a $C^{\infty }$ Whitney extension result, and a $C^{\infty }$ Lusin approximation result. Combined with previous work, this completes the study of Whitney extension and Lusin approximation for horizontal curves of class $C^m$, $C^{m,\omega }$, and $C^{\infty }$ in the Heisenberg group.

Let W be a finitely generated right-angled Coxeter group with group von Neumann algebra $L\left(W\right)$. We prove the following dichotomy: either $L\left(W\right)$ is strongly solid or W contains $Z\times F_2$ as a subgroup. This proves in particular strong solidity of $L\left(W\right)$ for all non-hyperbolic Coxeter groups that do not contain $Z\times F_2$.

We define and explicitly construct schemes evinced by generalized additive decompositions (GADs) of a given d-homogeneous polynomial F. We employ GADs to investigate the regularity of 0-dimensional schemes apolar to F, focusing on those satisfying some minimality conditions. We show that irredundant schemes to F need not be d-regular, unless they are evinced by special GADs of F. Instead, we prove that tangential decompositions of minimal length are always d-regular, as well as irredundant apolar schemes of length at most $2d+1$.