Annals of Pde最新文献

We prove local well-posedness for the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable layer (e.g., bedrock). Our result allows for the interface to touch the bottom, and hence applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer. We use this result in the companion paper Zlatoš [The 2D Muskat problem II: Stable regime small data singularity on the half-plane, preprint], to prove existence of finite time stable regime singularities in this model, including for arbitrarily small initial data. We do not require the interface and its derivatives to vanish at (pminfty) or be periodic, and even allow it to be (O(|x|^{1-})), which is an optimal bound on the power of growth. We also extend our result to the Muskat problem on the whole plane and on horizontal strips.

In this paper, we prove the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. This work builds on the framework developed in Häfner (Invent Math 223(3):1227–1406, 2021) for the study of the Einstein vacuum equations. We work in the generalized wave map and Lorenz gauge. The proof involves the analysis of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator on asymptotically flat spaces, which relies on recent advances in microlocal analysis and non-elliptic Fredholm theory developed in Vasy (Invent Math 194(2):381–513, 2013). The most delicate part of the proof is the description of the resolvent at low frequencies.

We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity (kappa > 0) and a force proportional to (sqrt{kappa}) on the flat (d)-torus (mathbb{T}^{d}) for (dgeq 2). We show the path-wise global well-posedness of the system and the existence of the invariant measures, and construct a random magnetohydrostatic (MHS) equilibrium (B(x)) in (H^{1}(mathbb{T}^{d})) with law (mathcal{D}(B)=mu_0) as a non-resistive limit (kappato 0) of statistically stationary solutions (B_{kappa}(x,t)). For (d=2), the measure (mu_0) does not concentrate on any compact subset in (H^{1}(mathbb{T}^{2})) with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium (B(x)) are almost surely not finite Fourier mode solutions.

In this article, we prove the global well-posedness in the critical Sobolev space (H_{rad}^2left(mathbb{R}^2right) times H_{rad}^1 left(mathbb{R}^2right)) for the radial time-like extremal hypersurface equation in (left(1+3right))- dimensional Minkowski space-time. This is achieved by deriving a new div-curl type lemma and combined it with energy and “momentum” balance law to get some space-time estimates of the nonlinearity.

In this paper, we investigate the ideal magnetohydrodynamics (MHD) equations on torus (mathbb{T}^d). For d = 3, we resolve the flexible part of Onsager-type conjecture for Elsässer energies of the ideal MHD equations. More precisely, for (beta < 1/3), we construct weak solutions ((u, b) in C^beta([0,T] times mathbb{T}^3)) with both the total energy dissipation and failure of cross helicity conservation. The key idea of the proof relies on a symmetry reduction that embeds the ideal MHD system into a 2(frac{1}{2})D Euler flow and the Newton-Nash iteration technique recently developed in  V. Giri (Invent Math 238:691–768, 2024). For d = 2, we show the non-uniqueness of Hölder-continuous weak solutions with non-trivial magnetic fields. Specifically, for (beta < 1/5), there exist infinitely many solutions ((u, b) in C^beta([0,T] times mathbb{T}^2)) with the same initial data while satisfying the total energy dissipation with non-vanishing velocity and magnetic fields. The new ingredient is developing a spatial-separation-driven iterative scheme that incorporates the magnetic field as a controlled perturbation within the convex integration framework for the velocity field, thereby providing sufficient oscillatory freedom for Nash-type perturbations in the 2D setting. As a byproduct, we prove that any Hölder-continuous Euler solution can be approximated by a sequence of C^β-weak solutions for the ideal MHD equations in the L^p-topology for (1le p < infty).

We study the elastic wave system in three spatial dimensions. For admissible harmonic elastic materials, we prove a desired low-regularity local well-posedness result for the corresponding elastic wave equations. For such materials, we can split the dynamics into the “divergence-part” and the “curl-part,” and each part satisfies a distinct coupled quasilinear wave system with respect to different acoustical metrics. Our main result is that the Sobolev norm (H^{3+}) of the “divergence-part” (the “faster-wave part”) and the ({H^{4 + }}) of the “curl-part” (the “slower-wave part”) can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption (H^{3+}) is optimal for the “divergence-part.” This marks the first favorable low-regularity local well-posedness result for a wave system with multiple wave speeds. Compared to the quasilinear wave equation, new difficulties arise from the multiple wave-speed nature of the system. Specifically, the acoustic metric (mathbf{g}) of the faster-wave depends on both the faster-wave and slower-wave parts. Additionally, the dynamics of the faster-wave “divergence-part” require higher regularity of the “curl-part”. In particular, the Ricci curvature associated with the faster-wave is one derivative rougher than that of the slower-wave dynamics.This phenomenon also appears in the compressible Euler equations (featuring multiple characteristic speeds) and is a major obstacle to obtaining low-regularity local well-posedness results for general quasilinear wave systems if the two parts do not exhibit strong decoupling properties or if the “curl-part” lacks the structure necessary for better regularity results. For the elastic wave system governing the dynamics of the admissible harmonic elastic materials, we report that we can overcome these difficulties. For this system, we exploit its geometric structures and find that the “divergence-part” and “curl-part” exhibit decoupling properties and both parts show regularity gains. Moreover, we prove that the “divergence-part” maintains to represent the faster-wave throughout the entire time of the existence of the solution, ensuring that the characteristic hypersurfaces of the faster-wave are spacelike with respect to the slower-wave. This implies a crucial coerciveness for the geometric cone-flux energy of the “curl-part” on such characteristic hypersurfaces of the “divergence-part.F We furthermore carefully address all these challenges through spacetime energy estimates, Strichartz estimates, frequency-localized decay estimates, and conformal energy estimates. In all these estimates, we also precisely trace the impact of the “curl-part” on the faster-wave dynamics and control the associated geometry via employing the vector field method and the Littlewood-Paley theory.

Building upon the idea in [Hou, arXiv:2404.09410 2024], we establish the stability of the type-I blowup with log correction for the complex Ginzburg-Landau equation. In the amplitude-phase representation, a generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel anisotropic modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose normalization conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. It enables us to establish the full stability of the blowup by enforcing vanishing conditions via the choice of normalization and using weighted energy estimates, for a non-variational problem. No topological argument or spectrum analysis is needed, opening up the possibility to tackle a wide range of type-I singularities. The log correction for the blowup rate is automatically inferred via the local normalization conditions, captured by the energy estimates and refined estimates of the modulation parameters.

We show that the number of quasinormal modes (QNM) for Schwarzschild and Schwarzschild–de Sitter black holes in a disc of radius r is bounded from below by cr³, proving that the recent upper bound by Jézéquel [Anal. PDE 17, 2024,] is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM, explaining the emergence of a distorted lattice and generalizing the lattice structure in strips described by Sá Barreto-Zworski [Math. Res. Lett. 4, 1997] (see Fig. 1). As a by-product we obtain an exponentially accurate Bohr–Sommerfeld quantization rule for one dimensional problems. The resulting description of QNM allows their accurate evaluation “deep in the complex” where numerical methods break down due to pseudospectral effects (see Fig. 2).

In this article, we investigate the Hölder regularity of the fractional (p)-Laplace equation of the form ((-Delta_p)^s u=f) where (p > 1, sin (0, 1)) and (fin L^infty_{rm loc}(Omega)). Specifically, we prove that (uin C^{0, gamma_circ}_{rm loc}(Omega)) for (gamma_circ=min{1, frac{sp}{p-1}}), provided that (frac{sp}{p-1}neq 1). In particular, it shows that (u) is locally Lipschitz for (frac{sp}{p-1} > 1). Moreover, we show that for (frac{sp}{p-1}=1), the solution is locally Lipschitz, provided that (f) is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.

In this work, we consider the relativistic Vlasov-Maxwell system, linearized around a spatially homogeneous equilibrium, set in the whole space (mathbb{R}^3 times mathbb{R}^3). The equilibrium is assumed to belong to a class of radial, smooth, rapidly decaying functions. Under appropriate conditions on the initial data, we prove algebraic decay (of dispersive nature) for the electromagnetic field. For the electric scalar potential, the leading behavior is driven by a dispersive wave packet with non-degenerate phase and compactly supported amplitude, while for the magnetic vector potential, it is driven by a wave packet whose phase behaves globally like the one of Klein-Gordon and the amplitude has unbounded support.