Analysis & PDE最新文献

We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues for the Dirichlet magnetic Laplacian with a nonuniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit, which also corresponds to a large magnetic field limit and is valid under a geometric assumption on the curvature of the discontinuity curve.

We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker–Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global-in-time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy, and barrier function methods.

The PDE approach developed earlier by the first three authors for $L^{\infty }$ estimates for fully nonlinear equations on Kähler manifolds is shown to apply as well to Monge–Ampère and Hessian equations on nef classes. In particular, one obtains a new proof of the estimates of Boucksom, Eyssidieux, Guedj and Zeriahi (2010) and Fu, Guo and Song (2020) for the Monge–Ampère equation, together with their generalization to Hessian equations.

We represent minimal upper gradients of Newtonian functions, in the range $1�\le �p�<�\infty$, by maximal directional derivatives along “generic” curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of $p$-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along $p$-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules.

The arising $p$-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger’s structure in the presence of a Poincaré inequality. In particular, it exists whenever the space is metrically doubling. It is moreover compatible with, and gives a geometric interpretation of, Gigli’s abstract differentiable structure, whenever it exists. The $p$-weak charts give rise to a finite-dimensional $p$-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.

We study singularity formation for the quadratic wave equation in the energy supercritical case, i.e., for $d�\ge 7$. We find in closed form a new, nontrivial, radial, self-similar blow-up solution $u^{\ast }$ which exists for all $d�\ge 7$. For $d�=9$, we study the stability of $u^{\ast }$ without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via $u^{\ast }$. In similarity coordinates, this family represents a codimension-1 Lipschitz manifold modulo translation symmetries. The stability analysis relies on delicate spectral analysis for a non-self-adjoint operator. In addition, in $d�=7$ and $d�=9$, we prove nonradial stability of the well-known ODE blow-up solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.

We prove necessary density conditions for sampling in spectral subspaces of a second-order uniformly elliptic differential operator on ${ℝ}^d$ with slowly oscillating symbol. For constant-coefficient operators, these are precisely Landau’s necessary density conditions for bandlimited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even exists. Our results prove the existence of a suitable critical sampling density and compute it in terms of the geometry defined by the elliptic operator. In dimension $d�=1$, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and we obtain a new critical density for variable bandwidth. The methods are a combination of the spectral theory and the regularity theory of elliptic partial differential operators, some elements of limit operators, certain compactifications of ${ℝ}^d$, and the theory of reproducing kernel Hilbert spaces.

We prove Strichartz inequalities for the Schrödinger equation and the wave equation with multiplicative noise on a two-dimensional manifold. This relies on the Anderson Hamiltonian described using high-order paracontrolled calculus. As an application, it gives a low-regularity solution theory for the associated nonlinear equations.

The paper is devoted to a comprehensive study of smoothness of inertial manifolds (IMs) for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,𝜀}$-regularity for such manifolds (for some positive, but small $𝜀$). Nevertheless, as shown in the paper, under natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n�\in �\mathbb{N}$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the $C^{1,𝜀}$-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.

We give a few explicit formulas regarding the radiation fields of linear free waves. We then apply these formulas on the channel-of-energy theory. We characterize all the radial weakly nonradiative solutions in all dimensions and give a few new exterior energy estimates.

We consider variational principles related to V. I. Arnold’s stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the Navier–Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen’s vortex, providing a new stability proof with stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex.