Annals of Pde最新文献

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.

In this paper, we seek to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result on the construction of nontrivial global solutions to a general system of quasilinear wave equations. Assuming that a global solution to the geometric reduced system exists and satisfies several well-chosen pointwise estimates, we find a matching exact global solution to the original wave equations. Such a conditional result is then applied to two types of equations which are of great interest. One is John’s counterexamples (Box u=u_t^2) or (Box u=u_t u_{tt}), and the other is the 3D compressible Euler equations with no vorticity. We explicitly construct global solutions to the corresponding geometric reduced systems and show that these global solutions satisfy the required pointwise bounds. As a result, there exists a large family of nontrivial global solutions to these two types of equations.

In this paper, we prove the first existence result of weak solutions to the 3D Euler equation with initial vorticity concentrated in a circle and velocity field in (C([0,T],L^{2^-})). The energy becomes finite and decreasing for positive times, with vorticity concentrated in a ring that thickens and moves in the direction of the symmetry axis. With our approach, there is no need to mollify the initial data or to rescale the time variable. We overcome the singularity of the initial data by applying convex integration within the appropriate time-weighted space.

We consider the Cauchy problem for the full free boundary Euler equations in 3d with an initial small velocity of size (O(varepsilon_0)), in a moving domain which is initially an (O(varepsilon_0)) perturbation of a flat interface. We assume that the initial vorticity is of size (O(varepsilon_1)) and prove a regularity result up to times of the order (varepsilon_1^{-1+}), independent of ({varepsilon _0}). A key part of our proof is a normal form type argument for the vorticity equation; this needs to be performed in the full three dimensional domain and is necessary to effectively remove the irrotational components from the quadratic stretching terms and uniformly control the vorticity. Another difficulty is to obtain sharp decay for the irrotational component of the velocity and the interface; to do this we perform a dispersive analysis on the boundary equations, which are forced by a singular contribution from the rotational component of the velocity. As a corollary of our result, when ({varepsilon _1}) goes to zero we recover the celebrated global regularity results of Wu (Invent. Math. 2012) and Germain, Masmoudi and Shatah (Ann. of Math. 2013) in the irrotational case.

Given a regular solution (mathbf{g}_0) of the Einstein-null dusts system without restriction on the number of dusts, we construct families of solutions ((mathbf{g}_lambda)_{lambdain(0,1]}) of the Einstein vacuum equations such that (mathbf{g}_lambda-mathbf{g}_0) and (partial(mathbf{g}_lambda-mathbf{g}_0)) converges respectively strongly and weakly to 0 when (lambdato0). Our construction, based on a multiphase geometric optics ansatz, thus extends the validity of the reverse Burnett conjecture without symmetry to a large class of massless kinetic spacetimes. In order to deal with the finite but arbitrary number of direction of oscillations we work in a generalised wave gauge and control precisely the self-interaction of each wave but also the interaction of waves propagating in different null directions, relying crucially on the non-linear structure of the Einstein vacuum equations. We also provide the construction of oscillating initial data solving the vacuum constraint equations and which are consistent with the spacetime ansatz.