Annals of PDE
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Conditional Stable Soliton Resolution for a Semi-linear Skyrme Equation
Annals of PDE - Tập 5 - Trang 1-59 - 2019
We study a semi-linear version of the Skyrme system due to Adkins and Nappi. The objects in this system are maps from $$(1+3)$$-dimensional Minkowski space into the 3-sphere and 1-forms on $$\mathbb {R}^{1+3}$$, coupled via a Lagrangian action. Under a co-rotational symmetry reduction we establish the existence, uniqueness, and unconditional asymptotic stability of a family of stationary solutions......
Emergence of Apparent Horizon in Gravitational Collapse
Annals of PDE - Tập 6 - Trang 1-89 - 2020
We solve Einstein vacuum equations in a spacetime region up to the “center” of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smooth (except at that point). In the proof we employ a scale critical trapped surfa......
On convergence of approximate solutions to the compressible Euler systemAbstract We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit. We show that such a sequence either (i) converges strongly in the energy norm, or (ii) the limit is no... ...
Annals of PDE - Tập 6 Số 2 - 2020
Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations
Annals of PDE - Tập 5 - Trang 1-51 - 2019
For all $$\epsilon >0$$, we prove the existence of finite-energy strong solutions to the axi-symmetric 3D Euler equations on the domains $$ \{(x,y,z)\in {\mathbb {R}}^3: (1+\epsilon |z|)^2\le x^2+y^2\}$$ which become singular in finite time. The solutions we construct have bounded vorticity before a certain time when the vorticity becomes unbounded. We further show that solutions with 0 swirl are ......
The Linear Stability of the Schwarzschild Solution to Gravitational Perturbations in the Generalised Wave Gauge
Annals of PDE - Tập 5 - Trang 1-92 - 2019
We prove in this paper that the Schwarzschild family of black holes are linearly stable as a family of solutions to the system of equations that result from expressing the Einstein vacuum equations in a generalised wave gauge. In particular we improve on our recent work (Johnson in The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge, arXi......
Non-local Functionals Related to the Total Variation and Connections with Image Processing
Annals of PDE - Tập 4 - Trang 1-77 - 2018
We present new results concerning the approximation of the total variation,
$$\int _{\Omega } |\nabla u|$$
, of a function u by non-local, non-convex functionals of the form
......
Localized Mixing Zone for Muskat Bubbles and Turned Interfaces
Annals of PDE - Tập 8 - Trang 1-50 - 2022
We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh–Taylor and smoothness breakdown exhibited in (Castro et al. in Arch Ration Mech Anal 208(......
Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier–Stokes Equations with Vacuum
Annals of PDE - - 2019
Dynamic Stability for Steady Prandtl Solutions
Annals of PDE - Tập 9 - Trang 1-33 - 2023
By establishing an invariant set (1.11) for the Prandtl equation in Crocco transformation, we prove the orbital and asymptotic stability of Blasius-like steady states against Oleinik’s monotone solutions.
The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity
Annals of PDE - - 2022
Tổng số: 109
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