Archive for Rational Mechanics and Analysis最新文献

Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system

We consider the critical mass case (int _{{mathbb {R}}^2} u_0(x), textrm{d}x = 8pi ), which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function (u_0^*) with mass (8pi ) such that for any initial condition (u_0) sufficiently close to (u_0^*) and mass (8pi ), the solution u(x, t) of ((*)) is globally defined and blows-up in infinite time. As (trightarrow +infty ) it has the approximate profile

where (lambda (t) approx frac{c}{sqrt{log t}}), (xi (t)rightarrow q) for some (c>0) and (qin {mathbb {R}}^2). This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).

We are interested in the regularity of weak solutions u to the elliptic equation in divergence form as in (1.1), and more precisely in their local boundedness and their local Lipschitz continuity under general growth conditions, the so called (p,!q)-growth conditions, as in (1.2) and (1.3) below. We found a unique set of assumptions to get all of these regularity properties at the same time; in the meantime we also found the way to treat a more general context, with explicit dependence on (left( x,uright) ), in addition to the gradient variable (xi =Du). These aspects require particular attention, due to the (p,!q)-context, with some differences and new difficulties compared to the standard case (p=q).

We consider weak solutions of the inhomogeneous non-cutoff Boltzmann equation in a bounded domain with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. When the mass, energy and entropy densities are bounded above, and the mass density is bounded away from a vacuum, we obtain an estimate of the (L^infty ) norm of the solution depending on the macroscopic bounds on these hydrodynamic quantities only. This is a regularization effect in the sense that the initial data is not required to be bounded. We present a proof based on variational ideas, which is fundamentally different to the proof that was previously known for the equation in periodic spatial domains.

We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order (alpha ), and the fractal Burgers equation of order (beta ), where (alpha , beta in [0,1)), and the Whitham equation. For all (alpha , beta in [0,1)), we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a “shock-like” singularity. Moreover, we provide an asymptotic description of the blow-up. To the best of our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range (alpha in [2/3, 1)), as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range (beta in [2/3, 1)). Our construction is based on modulation theory, where the well-known smooth self-similar solutions to the inviscid Burgers equation are used as profiles. A somewhat amusing point is that the profiles that are less stable under initial data perturbations (in that the number of unstable directions is larger) are more stable under perturbations of the equation (in that higher order dispersive and/or dissipative terms are allowed) due to their slower rates of concentration. Another innovation of this article, which may be of independent interest, is the development of a streamlined weighted (L^{2})-based approach (in lieu of the characteristic method) for establishing the sharp spatial behavior of the solution in self-similar variables, which leads to the sharp Hölder regularity of the solution up to the blow-up time.

We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (in: International mathematical congress, Heidelberg, 1904; see Gesammelte Abhandlungen II, 1961) and Batchelor (J Fluid Mech 1:177–190, 1956), any Euler solution arising in this limit and consisting of a single “eddy” must have constant vorticity. Feynman and Lagerstrom (in: Proceedings of IX international congress on applied mechanics, 1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. In the case of the disc, the choice—known to Batchelor (1956) and Wood (J Fluid Mech 2:77–87, 1957)—is explicit in terms of the slip forcing. For domains with non-constant curvature, Feynman and Lagerstrom give an approximate formula for the choice which is in fact only implicitly defined and must be determined together with the boundary layer profile. We show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. Due to the quasilinear coupling between the solution and the selected vorticity, we devise a delicate iteration scheme coupled with a high-order energy method that captures and controls the implicit selection mechanism.

Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a different approach by maximising directly the dispersive part of the energy functional, while keeping the remaining nonlinear terms fixed with an Orlicz-space constraint. This method is, to the best of our knowledge new in the setting of water waves. The constructed solutions are bell-shaped in the sense that they are even, one-sided monotone, and attain their maximum at the origin. The method initially considers weaker solutions than in earlier works, and is not limited to small waves: a family of solutions is obtained, along which the dispersive energy is continuous and increasing. In general, our construction admits more than one solution for each energy level, and waves with the same energy level may have different heights. Although a transformation in the construction hinders us from concluding the family with an extreme wave, we give a quantitative proof that the set reaches ‘large’ or ‘intermediate-sized’ waves.

There are many interesting dynamical systems in which degenerate invariant tori appear. We give conditions under which these degenerate tori have stable and unstable invariant manifolds, with stable and unstable directions having arbitrary finite dimension. The setting in which the dimension is larger than one was not previously considered and is technically more involved because in such case the invariant manifolds do not have, in general, polynomial approximations. As an example, we apply our theorem to prove that there are motions in the ((n+2))-body problem in which the distances among the first n bodies remain bounded for all time, while the relative distances between the first n-bodies and the last two and the distances between the last bodies tend to infinity, when time goes to infinity. Moreover, we prove that the final motion of the first n bodies corresponds to a KAM torus of the n-body problem.