We provide a unified viewpoint on two ill-posedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi-Mizohata condition. Our approach is based on a robust energy-and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydrodynamics. Concretely, the main results in this paper concern strong ill-posedness of the Cauchy problem (e.g., nonexistence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schr & ouml;dinger-and KdV-type equations, including the Hunter-Smothers equation, K (m, n) models of Rosenau-Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of the combination of two effects: degenerate dispersion-which is a property of the principal term in the presence of degenerating coefficients - and the evolution of the amplitude governed by the Takeuchi-Mizohata condition - which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical L2-ill-posedness result by Mizohata for linear variable-coefficient Schr & ouml;dinger equations with failed Takeuchi-Mizohata condition.
We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in R-3, which converges to a plane with multiplicity 2 as time approaches infinity.
We identify a new way to divide the delta-neighborhood of surfaces M C DI83 into a finitely overlapping collection of rectangular boxes S. We obtain a sharp (& ell;2, Lp) decoupling estimate using this decomposition for the sharp range of exponents 2 <= p <= 4. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line.
We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. This estimate is crucial in the integrability analysis mean-field interactions induced by a singular kernel. The tensorial operator arises from a reformulation of integrability. Additionally, for smooth periodic solutions of the reformulated system, we derive a stability result.
We introduce sparse versions of function spaces that are relevant to characterize the solutions of Euler equations without concentration. The standard Sobolev space $H^{-1}$ is given a sparse structure that allows to measure the degree of compactness of embeddings into $H^{-1}$ and provides new quantitative general criteria for $H^{-1}$-stability. Indices of sparseness are defined, and function spaces whose indices have prescribed decay are constructed, resulting in an improvement of the classical $H^{-1}$-stability results: sparse stability. The analysis relies on the introduction of sparse Riesz-Morrey-Tadmor spaces, that are characterized via maximal operators and new sparse domination theorems, together with extrapolation techniques. Our methods also yield improvements on recent results on the conservation of energy of physically realizable solutions of $2$D-Euler.
In this work, we study several properties of the normal Lebesgue trace of vector fields introduced by the second and third author in [22] in the context of the energy conservation for the Euler equations in Onsager-critical classes. Among other things, we prove that the normal Lebesgue trace satisfies the Gauss-Green identity and, by providing explicit counterexamples, that it is a notion sitting strictly between the distributional one for measure-divergence vector fields and the strong one for BV functions. These results are then applied to the study of the uniqueness of weak solutions for continuity equations on bounded domains, allowing to remove the assumption in [19] of global BV regularity up to the boundary, at least around the portion of the boundary where the characteristics exit the domain or are tangent. The proof relies on an explicit renormalization formula completely characterized by the boundary datum and the positive part of the normal Lebesgue trace. In the case when the characteristics enter the domain, a counterexample shows that achieving the normal trace in the Lebesgue sense is not enough to prevent non-uniqueness, and thus a BV assumption seems to be necessary to get uniqueness.
In this article we establish fine results on the boundary behavior of solutions to nonlocal equations in $C^{k,\gamma}$ domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like $v \asymp d^{s-1}$ and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients $v/d^{s-1}$, depending on the regularity of the domain and on the data of the problem. The results of this article will be important in a forthcoming work on nonlocal free boundary problems.
We consider here a fully discrete variant of the implicit variational scheme for mean curvature flow, see Almgren et al. (1993) and Luckhaus and Sturzenhecker (1995), in a setting where the flow is governed by a crystalline surface tension defined by the limit of pairwise interactions energy on the discrete grid. The algorithm is based on a new discrete distance from the evolving sets, which prevents the occurrence of the spatial drift and pinning phenomena identified in Misiats and Yip (2016) and Braides et al. (2010) in a similar discrete framework. We provide the first rigorous convergence result holding in any dimension, for any initial set and for a large class of purely crystalline anisotropies, in which the spatial discretization mesh can be of the same order or coarser than the time step.
The main goal of this paper is to generalize a part of the relationship between mean curvature and Harder-Narasimhan filtrations of holomorphic vector bundles to arbitrary polarized fibrations. More precisely, for a polarized family of complex projective manifolds, we establish lower bounds on a fibered version of Yang-Mills functionals in terms of the Harder-Narasimhan slopes of direct image sheaves associated with high tensor powers of the polarization. We discuss the optimality of these lower bounds and, as an application, provide an analytic characterisation of a fibered version of generic nefness. As another application, we refine the existent obstructions for finding metrics with constant horizontal mean curvature. The study of the semiclassical limit of Hermitian Yang-Mills functionals lies at the heart of our approach.
New, doubling proofs are given for the interior Hessian estimates of the special Lagrangian equation. These estimates were originally shown by Chen-Warren-Yuan in CPAM 2009 and Wang-Yuan in AJM 2014. This yields a higher codimension analogue of Korevaar's 1987 pointwise proof of the gradient estimate for minimal hypersurfaces, without using the Michael-Simon mean value inequality.
We develop potential theory for m-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to define the m-capacity and then showing the quasi-continuity of m-subharmonic functions. Thanks to this we derive other results parallel to those in pluripotential theory such as the equivalence between polar sets and negligible sets. The theory is then used to study the complex Hessian equation on compact Hermitian manifold with boundary, with the right hand side of the equation admitting a bounded subsolution. This is an extension of a recent result of Collins and Picard dealing with classical solutions.
We consider the dynamics of a 2D Bose gas with an interaction potential of the form $N^{2\beta-1}w(N^\beta\cdot)$ for $\beta\in (0,3/2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schr{\"o}dinger equation (NLS) may blow up in finite time. We show that to leading order, the $N$-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics.
We investigate existence and nonexistence of action ground states and nodal action ground states for the nonlinear Schr\"odinger equation on noncompact metric graphs with rather general boundary conditions. We first obtain abstract sufficient conditions for existence, typical of problems with lack of compactness, in terms of ``levels at infinity'' for the action functional associated with the problems. Then we analyze in detail two relevant classes of graphs. For noncompact graphs with finitely many edges, we detect purely topological sharp conditions preventing the existence of ground states or of nodal ground states. We also investigate analogous conditions of metrical nature. The negative results are complemented by several sufficient conditions to ensure existence, either of topological or metrical nature, or a combination of the two. For graphs with infinitely many edges, all bounded, we focus on periodic graphs and infinite trees. In these cases, our results completely describe the phenomenology. Furthermore, we study nodal domains and nodal sets of nodal ground states and we show that the situation on graphs can be totally different from that on domains of $\mathbb{R}^N$.
We present a novel approach to the analysis of regularity and decay for solutions of wave equations in a neighborhood of null infinity in asymptotically flat spacetimes of any dimension. The classes of metrics and wave type operators we consider near null infinity include those arising in nonlinear stability problems for Einstein's field equations in 1+3 dimensions. In a neighborhood of null infinity, in an appropriate compactification of the spacetime to a manifold with corners, the wave operators are of edge type at null infinity and totally characteristic at spacelike and future timelike infinity. On a corresponding scale of Sobolev spaces, we demonstrate how microlocal regularity propagates across or into null infinity via a sequence of radial sets. As an application, inspired by work of the second author with Baskin and Wunsch, we prove regularity and decay estimates for forward solutions of wave type equations on asymptotically flat spacetimes which are asymptotically homogeneous with respect to scaling in the forward timelike cone and have an appropriate structure at null infinity. These estimates are new even for the wave operator on Minkowski space. The results obtained here are also used as black boxes in a global theory of wave type equations on asymptotically flat and asymptotically stationary spacetimes developed by the first author.
We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes in general relativity. There are two distinguishing aspects of our approach. The first aspect is its dyadically localised nature: The nontrivial part of the analysis is reduced entirely to time-translation invariant $r^p$-weighted estimates, in the spirit of [DR09], to be applied on dyadic time-slabs which for large $r$ are outgoing. Global existence and decay then both immediately follow by elementary iteration on consecutive such time-slabs, without further global bootstrap. The second, and more fundamental, aspect is our direct use of a "blackbox" linear inhomogeneous energy estimate on exactly stationary metrics, together with a novel but elementary physical space top order identity that need not capture the structure of trapping and is robust to perturbation. In the specific example of Kerr black holes, the required linear inhomogeneous estimate can then be quoted directly from the literature [DRSR16], while the additional top order physical space identity can be shown easily in many cases (we include in the Appendix a proof for the Kerr case $\vert a \vert \ll M$, which can in fact be understood in this context simply as a perturbation of Schwarzschild). In particular, the approach circumvents the need either for producing a purely physical space identity capturing trapping or for a careful analysis of the commutation properties of frequency projections with the wave operator of time-dependent metrics.
We prove a dimension-free L-p(R-d) estimate, 1 < p < infinity, for the vector of higher-order maximal Riesz transforms in terms of the corresponding Riesz transforms. This implies a dimension-free L-p(R-d) estimate for the vector of maximal Riesz transforms in terms of the input function. We also give explicit estimates for the dependencies of the constants on p when the order is fixed. Analogous dimension-free estimates are also obtained for single higher-order Riesz transforms with an improved estimate of the constants.
We consider heat operators on a convex domain Ω, with a critically singular potential that diverges as the inverse square of the distance to the boundary of Ω. We establish a general boundary controllability result for such operators in all dimensions, in particular providing the first such result in more than one spatial dimension. The key step in the proof is a novel global Carleman estimate that captures both the appropriate boundary conditions and the H^1-energy for this problem. The estimate is derived by combining two intermediate Carleman inequalities with distinct and carefully constructed weights involving non-smooth powers of the boundary distance.
In 1994 Velazquez constructed a smooth \(O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velazquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Here we prove short time existence of Velazquez formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velazquez-Stolarski we therefore show that there exists a solution \(\{M_t^7\subset\R^8 \mid -t_0
We investigate maximal potential energy dissipation as a selection criterion for subsolutions (coarsegrained solutions) in the setting of the unstable Muskat problem. We show that both (a) imposing this criterion on the level of convex integration subsolutions and (b) the strategy of Otto based on a relaxation via minimizing movements lead to the same nonlocal conservation law. Our main result shows that this equation admits an entropy solution for unstable initial data with an analytic interface.
