Annals of Pde最新文献

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.

We consider the linear transport equations driven by an incompressible flow in dimensions (dge 3). For divergence-free vector fields (u in L^1_t W^{1,q}), the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class (L^infty _t L^p) when (frac{1}{p} + frac{1}{q} le 1). For such vector fields, we show that in the regime (frac{1}{p} + frac{1}{q} > 1), weak solutions are not unique in the class ( L^1_t L^p). One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

We show that the linearized Vlasov-Poisson equations around traveling wave-like non-homogeneous states near zero contain the full plasma echo mechanism, yielding Gevrey 3 as a critical stability class. Moreover, here Landau damping may persist despite blow-up: We construct a critical Gevrey regularity class in which the force field converges in (L^2). Thus, on the one hand, the physical phenomenon of Landau damping holds. On the other hand, the density diverges to infinity in Sobolev regularity. Hence, “strong damping” cannot hold.

We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant M and a sufficiently small parameter (varepsilon ), for generic initial data of size (varepsilon ), the flow is conjugated to an integrable flow up to an arbitrary small remainder of order (varepsilon ^{M+1}). This implies that for such initial data u(0) we control the Sobolev norm of the solution u(t) for time of order (varepsilon ^{-M}). Furthermore this property is locally stable: if v(0) is sufficiently close to u(0) (of order (varepsilon ^{3/2})) then the solution v(t) is also controled for time of order (varepsilon ^{-M}).

We study blowup solutions of the 6D energy critical heat equation (u_t=Delta u+|u|^{p-1}u) in ({mathbb {R}}^ntimes (0,T)). A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 456(2004):2957–2982, 2000). The dimension six is a border case whether a type II blowup can occur or not. Therefore the behavior of the solution is quite different from other cases. In fact, our solution behaves like

with (lambda (t)=(1+o(1))(T-t)^frac{5}{4}|log (T-t)|^{-frac{15}{8}}). Particularly the local energy defined by (E_{text {loc}}(u(t)) =frac{1}{2}Vert nabla u(t)Vert _{L^2(|x|<1)}^2-frac{1}{p+1}Vert u(t)Vert _{L^{p+1}(|x|<1)}^{p+1}) goes to (-infty ).

We present the first rigorous study of nonlinear wave equations on extremal black hole spacetimes without any symmetry assumptions on the solution. Specifically, we prove global existence with asymptotic blow-up for solutions to nonlinear wave equations satisfying the null condition on extremal Reissner–Nordström backgrounds. This result shows that the extremal horizon instability persists in model nonlinear theories. Our proof crucially relies on a new vector field method that allows us to obtain almost sharp decay estimates.