We consider the three-dimensional ideal MHD system on a domain Ω ' ⊂ℝ^3 with an open subset Γ of the boundary ∂Ω ' where we prescribe both u· and b· , while u·= b·=0 on ∂Ω ' ∖Γ . We prove boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with an initial state (u_0,b_0) achieves another state (u_1,b_1) in a finite time, where u_0,b_0,u_1,b_1 are arbitrary divergence-free vector fields satisfying impermeability boundary condition and which are extendible to vector fields with the same properties on any bounded domain obtained by extension of Ω ' via Γ . As a byproduct, we give the first local well-posedness proof for the incompressible ideal MHD system which does not use Elsasser variables and is thus applicable to any bounded domain with sufficient Sobolev regularity. We also provide a new, simpler control for the two-dimensional ideal MHD system.
Regularity of the Boltzmann equation, particularly in the presence of physical boundary conditions, heavily relies on the geometry of the boundaries. In the case of non-convex domains with specular reflection boundary conditions, the problem remained outstanding until recently due to the severe singularity of billiard trajectories near the grazing set, where the trajectory map is not differentiable. This challenge was addressed in Kim and Lee (Commun Pure Appl Math 77(4):2331-2386, 2024), where Cx,v12- H & ouml;lder regularity was proven. In this paper, we introduce a novel dynamical singular regime integration methodology to establish the optimal Cx,v12 regularity for the Boltzmann equation past a convex obstacle.
Regularity of the Boltzmann equation, particularly in the presence of physical boundary conditions, heavily relies on the geometry of the boundaries. In the case of non-convex domains with specular reflection boundary conditions, the problem remained outstanding until recently due to the severe singularity of billiard trajectories near the grazing set, where the trajectory map is not differentiable. This challenge was addressed in Kim and Lee (Commun Pure Appl Math 77(4):2331–2386, 2024), where C^1/2-_x,v Hölder regularity was proven. In this paper, we introduce a novel dynamical singular regime integration methodology to establish the optimal C^1/2_x,v regularity for the Boltzmann equation past a convex obstacle.
We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity κ > 0 and a force proportional to √(κ) on the flat d -torus 𝕋^d for d≥ 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(𝕋^d) with law 𝒟(B)=μ_0 as a non-resistive limit κ→ 0 of statistically stationary solutions B_κ(x,t) . For d=2 , the measure μ_0 does not concentrate on any compact subset in H^1(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.
This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets following the formation of curvature singularities due to the Kelvin-Helmholtz instability. Furthermore, they constitute plausible candidates for demonstrating non-uniqueness within the class of Delort’s weak solutions. The most challenging part of this paper is handling the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.
We are concerned with the low regularity of self-similar solutions of two-dimensional Riemann problems for the isentropic Euler system. We establish a general framework for the analysis of the local regularity of such solutions for a class of two-dimensional Riemann problems for the isentropic Euler system, which includes the regular shock reflection problem, the Prandtl reflection problem, the Lighthill diffraction problem, and the four-shock Riemann problem. We prove that the velocity is not in H^1 in the subsonic domain for the self-similar solutions of these problems in general. This indicates that the self-similar solutions of the Riemann problems with shocks for the isentropic Euler system are of much more complicated structure than those for the Euler system for potential flow; in particular, the velocity is not necessarily continuous in the subsonic domain. The proof is based on a regularization of the isentropic Euler system to derive the transport equation for the vorticity, a renormalization argument extended to the case of domains with boundary, and DiPerna-Lions-type commutator estimates.
In this paper, we prove the global well-posedness of the energy-critical nonlinear Schrödinger equations on the torus 𝕋^d for general dimensions. This result is new for dimensions d≥ 5 , extending previous results for d=3,4 [11, 23]. Compared to the cases d=3,4 , the regularity theory for higher d, developed in the underlying local well-posedness result [18], is less understood. In particular, stability theory and inverse inequalities, which are ingredients in [11, 23] and more generally in the widely used concentration compactness framework since [14], are too weak to be applied to higher dimensions. Our proof introduces a new strategy for addressing global well-posedness problems. Without relying on perturbation theory, we develop tools to analyze the concentration dynamics of the nonlinear flow. On the way, we show the formation of a nontrivial concentration.
We substantially improve in two scenarios the current state-of-the-art modulus of continuity for weak solutions to the N- dimensional, two-phase Stefan problem featuring a p- degenerate diffusion: for p=N≥ 3 , we sharpen it to ω(r) ≈exp (-c| ln r|^1/N); for p>max{2,N} , we derive an unexpected Hölder modulus.
Let p 2 . For any small enough r> max{p-1,1} and for any Λ > 1 there exists a Lipschitz function u and a bounded vectorfield f such that {[ div(|∇ u|^p-2∇ u) = div (f) in 𝔹^2; u=0 on ∂𝔹^2 ]. but ∫ _𝔹^2 |∇ u|^r ≰Λ∫ _𝔹^2 |f|^r/p-1. This disproves a conjecture by Iwaniec from 1983.
In this work we prove global well-posedness for the massive Maxwell-Dirac system in the Lorenz gauge in ℝ^1+3, for small, sufficiently smooth and decaying initial data, as well as modified scattering for the solutions. Heuristically we exploit the close connection between the massive Maxwell-Dirac and the wave-Klein-Gordon equations, while developing a novel approach which applies directly at the level of the Dirac equations. The modified scattering result follows from a precise description of the asymptotic behavior of the solutions inside the light cone, which we derive via the method of testing with wave packets of Ifrim-Tataru.
In this paper, we analyze a three-dimensional Nernst-Planck-Boussinesq (NPB) system that describes ionic electrodiffusion in an incompressible viscous fluid. This new model incorporates variational temperature and is forced by buoyancy force stemming from temperature and salinity fluctuations, enhancing its generality and realism. The electromigration term in the NPB system displays a complex nonlinear structure influenced by the reciprocal of the temperature that distinguishes its mathematical aspects from other electrodiffusion models studied in the literature. We address the global existence of weak solutions to the NPB system on the three-dimensional torus for large initial data. In addition, we study the long-time dynamics of these weak solutions and the associated relative entropies and establish their exponential decay in time to steady states.
We consider the cubic Schrödinger equation posed on a product space subject to a generic Diophantine condition. Our analysis shows that the small-amplitude solutions undergo modified scattering to an effective dynamics governed by some interactions that do not amplify the Sobolev norms. This is in sharp contrast with the infinite energy cascade scenario observed by Hani–Pausader–Tzvetkov–Visciglia in the absence of Diophantine conditions.
We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in H^s,0(ℝ^2) for s>3/4 and unconditional global well-posedness in the energy space. We also prove the global existence of perturbations with finite energy of non decaying smooth global solutions.
We consider a simplified extensible version of a dynamic free boundary problem for a thin filament with radius ϵ >0 immersed in 3D Stokes flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel type of angle-averaged Neumann-to-Dirichlet operator for the Stokes equations, and much of the difficulty in the analysis lies in understanding this operator. Here we show that the main part of this angle-averaged NtD map about a closed, curved filament is the corresponding operator about a straight filament, for which we can derive an explicit symbol. Remainder terms due to curvature are lower order with respect to regularity or size in ϵ . Using this operator decomposition, it is then possible to show that the simplified free boundary evolution is a third-order parabolic equation and is locally well posed. This establishes a more complete mathematical foundation for the myriad computational results based on slender body approximations for thin immersed elastic structures.
Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$ for \textit{all} $\ell>1$ in physical space dimension $d=2,3$ and for $\ell>1$ close to 1 in higher dimensions. This work is a crucial step toward solving the long-standing problem: finite time blow-up of the supercritical defocusing nonlinear wave equation.
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 ±∞ 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.
Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system * { u_t = Δ u - ∇· (u ∇ v) in ℝ^2× (0,T), v = (-Δ _ℝ^2)^-1 u := 1/2π∫ _ℝ^2log1/|x-z| u(z,t) dz, u(· ,0) = u_0^⋆≥ 0 in ℝ^2. . We show that there exists ε >0 such that for any m satisfying 8π
In this paper, we establish the stability of the quasineutral limit for the ionic Vlasov-Poisson system under perturbations exponentially small in Wasserstein sense. Notably, we emphasize that exponential smallness is a necessary condition in the electron case, as the presence of instabilities makes polynomial smallness insufficient. The study's quantitative nature introduces unique challenges, primarily arising from the exponential Poisson coupling. These challenges necessitate careful optimization at every step of the proof, whether it be in refining estimates or in the overall approach. Within this paper, we introduce novel tools and approaches to address these challenges. Specifically, we enhance the existing theory concerning the growth of characteristics in Vlasov systems featuring nonlinear couplings. Additionally, we combine stability estimates using kinetic-Wasserstein distances with improved regularity bounds on the elliptic coupling. In the course of demonstrating our central result, we also enhance the moment assumptions associated with the well-posedness of the ionic Vlasov-Poisson system.
We study, fully microlocally, the propagation of massive waves on the octagonal compactification 𝕆=[ℝ^1,d;ℐ;1/2] of asymptotically Minkowski spacetime, which allows a detailed analysis both at timelike and spacelike infinity (as previously investigated using Parenti–Shubin–Melrose’s sc-calculus) and, more novelly, at null infinity, denoted ℐ . The analysis is closely related to Hintz–Vasy’s recent analysis of massless wave propagation at null infinity using the “e,b-calculus” on 𝕆 . We prove several elementary corollaries regarding the Klein–Gordon IVP. Our main technical tool is a fully symbolic pseudodifferential calculus, Ψ _de,sc(𝕆) , the “de,sc-calculus” on 𝕆 . The ‘de’ refers to the structure (“double edge”) of the calculus at null infinity, and the ‘sc’ refers to the structure (“scattering”) at the other boundary faces. We relate this structure to the hyperbolic coordinates used in other studies of the Klein–Gordon equation. Unlike hyperbolic coordinates, the de,sc- boundary fibration structure is Poincaré invariant.
