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Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow-up in the norm at some finite time occurs only if a certain quotient involving and its Newtonian potential concentrates near zero, which implies blow-up in more standard norms, such as the norm. This quotient is shown to be always less than a universal constant, suggesting that the problem of regularity for the Landau equation is in some sense critical.
The bounds are obtained using the comparison principle both for the Landau equation and for the associated mass function. In particular, the method provides long-time existence results for a modified version of the Landau equation with Coulomb potential, recently introduced by Krieger and Strain.
In the author’s previous work, it has been shown that solutions of Maxwell–Klein–Gordon equations in possess some form of global strong decay properties with data bounded in some weighted energy space. In this paper, we prove pointwise decay estimates for the solutions for the case when the initial data are merely small on the scalar field but can be arbitrarily large on the Maxwell field. This extends the previous result of Lindblad and Sterbenz, in which smallness was assumed both for the scalar field and the Maxwell field.
We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric -tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal -tensors. We work with compact simple manifolds, but several of our results apply to nontrapping manifolds with strictly convex boundary.
We develop a new method of proving vector-valued estimates in harmonic analysis, which we call “the helicoidal method”. As a consequence of it, we are able to give affirmative answers to several questions that have been circulating for some time. In particular, we show that the tensor product between the bilinear Hilbert transform and a paraproduct satisfies the same estimates as the itself, solving completely a problem introduced by Muscalu et al. (ActaMath.193:2 (2004), 269–296). Then, we prove that for “locally exponents” the corresponding vector-valued satisfies (again) the same estimates as the itself. Before the present work there was not even a single example of such exponents.
Finally, we prove a biparameter Leibniz rule in mixed norm spaces, answering a question of Kenig in nonlinear dispersive PDE.
We show that any amenable von Neumann subalgebra of any free Araki–Woods factor that is globally invariant under the modular automorphism group of the free quasifree state is necessarily contained in the almost periodic free summand.
We consider the global regularity problem for defocusing nonlinear wave systems
on Minkowski spacetime with d’Alembertian , where the field is vector-valued, and is a smooth potential which is positive and homogeneous of order outside of the unit ball for some . This generalises the scalar defocusing nonlinear wave (NLW) equation, in which and . It is well known that in the energy-subcritical and energy-critical cases when or and , one has global existence of smooth solutions from arbitrary smooth initial data , at least for dimensions . We study the supercritical case where and . We show that in this case, there exists a smooth potential for some sufficiently large (in fact we can take ), positive and homogeneous of order outside of the unit ball, and a smooth choice of initial data for which the solution develops a finite-time singularity. In fact the solution is discretely self-similar in a backwards light cone. The basic strategy is to first select the mass and energy densities of , then itself, and then finally design the potential in order to solve the required equation. The Nash embedding theorem is used in the second step, explaining the need to take relatively large.
We construct for every integer a complex manifold of dimension which is exhausted by an increasing sequence of biholomorphic images of (i.e., a long ), but does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new holomorphic invariants of a complex manifold , the stable core and the strongly stable core, which are based on the long-term behavior of hulls of compact sets with respect to an exhaustion of . We show that every compact polynomially convex set such that is the strongly stable core of a long ; in particular, holomorphically nonequivalent sets give rise to nonequivalent long ’s. Furthermore, for every open set there exists a long whose stable core is dense in . It follows that for any there is a continuum of pairwise nonequivalent long ’s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long-standing open problems.