Annals of Pde最新文献

We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, possibly with a small gravity constant, that feature a splash singularity. More precisely, in the solutions we construct the interface is a (mathcal {C}^{2,alpha }) smooth curve that intersects itself at one point, and the vorticity density on the interface is of class (mathcal {C}^alpha ). The proof consists in perturbing Crapper’s family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid. These estimates will also be applied to interface evolution problems in a forthcoming paper.

In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas’ GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold ({mathcal {H}}) closely related to Solomon’s space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with (C^{1,alpha }) regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen’s theorem on the existence of (C^{1,alpha }) geodesics in the space of Kähler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM [26] and special Lagrangians in Landau–Ginzburg models [27].

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models

The orbital stability of kinks under general assumptions on the potential W is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential W for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the (P(phi )_2) theories and the double sine-Gordon theory.

This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equations. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao [4]. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions bounded by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality.

In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an (L^2)-dense set of Hölder continuous initial data in the class of Hölder continuous admissible weak solutions for all exponents below the Onsager-critical 1/3. Along the way, and more importantly, we identify a natural condition on “blow-up” of the associated subsolution, which acts as the signature of the non-uniqueness mechanism. This improves previous results on non-uniqueness obtained in (Daneri in Comm. Math. Phys. 329(2):745–786, 2014; Daneri and Székelyhidi in Arch. Rat. Mech. Anal. 224: 471–514, 2017) and generalizes (Buckmaster et al. in Comm. Pure Appl. Math. 72(2):229–274, 2018).

We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in (L^2). Moreover, we prove that the solution exists globally in time under a smallness assumption.

We consider the SQG equation with dissipation given by a fractional Laplacian of order (alpha <frac{1}{2}). We introduce a notion of suitable weak solution, which exists for every (L^2) initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most (frac{1}{2alpha } left( frac{1+alpha }{alpha } (1-2alpha ) + 2right) ).

Gajic–Warnick [8] have recently proposed a definition of scattering resonances based on Gevrey-2 regularity at infinity and introduced a new class of potentials for which resonances can be defined. We show that standard methods based on complex scaling apply to a larger class of potentials and provide a definition of resonances in wider angles.

We study patch solutions of a family of transport equations given by a parameter (alpha ), (0< alpha <2), with the cases (alpha =0) and (alpha =1) corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for (H^{2}) patches in the half-space setting for (0<alpha < 1/3), allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of (alpha ) for which finite time singularities have been shown in Kiselev et al. (Commun Pure Appl Math 70(7):1253–1315, 2017) and Kiselev et al. (Ann Math 3:909–948, 2016). Second, we establish that patches remain regular for (0<alpha <2) as long as the arc-chord condition and the regularity of order (C^{1+delta }) for (delta >alpha /2) are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in Córdoba et al. (Proc Natl Acad Sci USA 102:5949–5952, 2005) and Scott and Dritschel (Phys Rev Lett 112:144505, 2014) for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in Gancedo (Adv Math 217(6):2569–2598, 2008) and in Chae et al. (Commun Pure Appl Math 65(8):1037–1066, 2012), giving local existence for patches in (H^{2}) for (0<alpha < 1) and in (H^3) for (1<alpha <2).

We derive the global dynamic properties of the mMKG system (Maxwell coupled with a massive Klein-Gordon scalar field) with a general class of data, in particular, for Maxwell field of arbitrary size, and by a gauge independent method. Due to the critical slow decay expected for the Maxwell field, the scalar field exhibits a loss of decay at the causal infinities within an outgoing null cone. To overcome the difficulty caused by such loss in the energy propagation, we uncover a hidden cancellation contributed by the Maxwell equation, which enables us to obtain the sharp control of the Maxwell field under a rather low regularity assumption on data. Our method can be applied to other physical field equations, such as the Einstein equations for which a similar cancellation structure can be observed.