We investigate the long time behaviour of the Yang-Mills heat flow on the bundle R4 & times;SU(2) . Waldron (see Waldron, A.: Long-time existence for Yang-Mills flow. Invent. Math. 4, 217, 1069-1147 (2019)) proved global existence and smoothness of the flow on closed 4-manifolds, leaving open the issue of the behaviour in infinite time. We exhibit two types of long-time bubbling: first we construct an initial data and a globally defined solution which blows-up in infinite time at a given point in R4 . Second, we prove the existence of bubble-tower solutions, also in infinite time. This answers the basic dynamical properties of the heat flow of Yang-Mills connection in the critical dimension 4 and shows in particular that in general one cannot expect that this gradient flow converges to a Yang-Mills connection. We emphasize that we do not assume for the first result any symmetry assumption; whereas the second result on the existence of the bubble-tower is in the SO(4)-equivariant class, but nevertheless new.
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.
In this article we consider one-dimensional scalar quasilinear Klein–Gordon equations with general nonlinearities, on both ℝ and 𝕋 . By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions. Our main result asserts that solutions with small initial data of size ϵ persist on the improved cubic timescale |t| ≲ϵ ^-2 and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case of ℝ , we are further able to use dispersion in order to extend the lifespan to ϵ ^-4 . This generalizes earlier results obtained by Delort [8] in the semilinear case.
We consider the kinematic dynamo equations for a passive vector in ℳ×𝕋⊆ℝ^2 ×𝕋 describing the evolution of a magnetic field with resistivity ε>0, that is transported by a given velocity field. For a broad class of C^3 velocity fields with helical geometry, we establish the existence of solutions that exhibit exponential growth over time. We construct an unstable eigenmode via detailed resolvent estimates of the corresponding linear operator, which we carry out by introducing suitable Green's functions that accurately approximate the local behaviour of the true system. This approach yields an explicit asymptotic expression for the growing mode, providing a sharp description of the instability mechanism. We first derive the results with ℳ=ℝ^2 for a large class of velocity fields that includes finite energy examples. We then consider the case of domains with boundary, where ℳ×𝕋 denotes a periodic cylinder, annular cylinder, or the exterior of a cylinder, with the boundary conditions of perfectly conducting walls. Our results offer a rigorous and sharp mathematical justification for the physically conjectured process by which helical flows can sustain magnetic field generation in the Ponomarenko dynamo, with growth rate of order ε^1/3.
