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Abstract: Abstract In this paper, we investigate the dynamics of solutions of the Muskat equation with initial interface consisting of multiple corners allowing for linear growth at infinity. Specifically, we prove that if the initial data contains a finite set of small corners then we can find a precise description of the solution showing how these corners desingularize and move at the same time. At the analytical level, we are solving a small data critical problem which requires renormalization. This is accomplished using a nonlinear change of variables which serves as a logarithmic correction and accurately describes the motion of the corners during the evolution. PubDate: 2024-08-26

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Abstract: Abstract We consider the global evolution problem for Einstein’s field equations in the near-Minkowski regime and study the long-time dynamics of a massive scalar field evolving under its own gravitational field. We establish the existence of a globally hyperbolic Cauchy development associated with any initial data set that is sufficiently close to a data set in Minkowski spacetime. In addition to applying to massive fields, our theory allows us to cover metrics with slow decay in space. The strategy of proof, proposed here and referred to as the Euclidean-Hyperboloidal Foliation Method, applies, more generally, to nonlinear systems of coupled wave and Klein-Gordon equations. It is based on a spacetime foliation defined by merging together asymptotically Euclidean hypersurfaces (covering spacelike infinity) and asymptotically hyperboloidal hypersurfaces (covering timelike infinity). A transition domain (reaching null infinity) limited by two asymptotic light cones is introduced in order to realize this merging. On the one hand, we exhibit a boost-rotation hierarchy property (as we call it) which is associated with Minkowski’s Killing fields and is enjoyed by commutators of curved wave operators and, on the other hand, we exhibit a metric hierarchy property (as we call it) enjoyed by components of Einstein’s field equations in frames associated with our Euclidean-hyperboloidal foliation. The core of the argument is, on the one hand, the derivation of novel integral and pointwise estimates which lead us to almost sharp decay properties (at timelike, null, and spacelike infinity) and, on the other hand, the control of the (quasi-linear and semi-linear) coupling between the geometric and matter parts of the Einstein equations. PubDate: 2024-08-07

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Abstract: Abstract This paper investigates the global dynamics of the apparent horizon. We present an approach to establish its existence and its long-term behaviors. Our apparent horizon is constructed by solving the marginally outer trapped surface (MOTS) along each incoming null hypersurface. Based on the nonlinear hyperbolic estimates established in [21] by Klainerman-Szeftel under polarized axial symmetry, we prove that the corresponding apparent horizon is smooth, asymptotically null and converging to the event horizon eventually. To further address the local achronality of the apparent horizon, a new concept, called the null comparison principle, is introduced in this paper. For three typical scenarios of gravitational collapse, our null comparison principle is tested and verified, which guarantees that the apparent horizon must be piecewise spacelike or piecewise null. In addition, we also validate and provide new proofs for several physical laws along the apparent horizon. PubDate: 2024-08-03

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Abstract: Abstract We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near equilibrium nor self-similar, to the equation, and prove that the well/ill-posedness threshold in \(H^{s}\) Sobolev space is exactly at regularity \(s=1\) , despite the fact that the equation is scale invariant at \( s=\frac{1}{2}\) . PubDate: 2024-07-24

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Abstract: Abstract In this paper, we study the 2D free boundary incompressible Euler equations with surface tension, where the fluid domain is periodic in \(x_1\) , and has finite depth. We construct initial data with a flat free boundary and arbitrarily small velocity, such that the gradient of vorticity grows at least double-exponentially for all times during the lifespan of the associated solution. This work generalizes the celebrated result by Kiselev–Šverák [17] to the free boundary setting. The free boundary introduces some major challenges in the proof due to the deformation of the fluid domain and the fact that the velocity field cannot be reconstructed from the vorticity using the Biot-Savart law. We overcome these issues by deriving uniform-in-time control on the free boundary and obtaining pointwise estimates on an approximate Biot-Savart law. PubDate: 2024-07-05

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Abstract: Abstract In the paper by Klainerman, Rodnianski and Tao [7], they give a physical space proof to a classical result of Klainerman and Machedon [3] for the bilinear space-time estimates of null forms. In this paper, we shall give an alternative and very simple physical space proof of the same bilinear estimates by applying div-curl type lemma of Zhou [14] and Wang and Zhou [12, 13]. We have only attained the limited goal of proving the bilinear estimates for the dyadic piece of the solution. Summing up the dyadic parts leads to the bilinear estimates with a Besov loss. As far as we know, the later development of wave maps [1, 2, 8–11], and the proof of bounded curvature theorem [5, 6] rely on basic ideas of Klainerman and Machedon [3] as well as Klainerman, Rodnianski and Tao [7]. PubDate: 2024-06-18 DOI: 10.1007/s40818-024-00176-x

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Abstract: Abstract We consider the two-dimensional stationary Navier–Stokes equations on the whole plane \(\mathbb {R}^2\) . In the higher-dimensional cases \(\mathbb {R}^n\) with \(n \geqslant 3\) , the well-posedness and ill-posedness in scaling critical spaces are well-investigated by numerous papers. However, the corresponding problem in the two-dimensional whole plane case has been known as an open problem due to inherent difficulties of two-dimensional analysis. The aim of this paper is to address this issue and solve it negatively. More precisely, we prove the ill-posedness in the scaling critical Besov spaces based on \(L^p(\mathbb {R}^2)\) for all \(1 \leqslant p \leqslant 2\) in the sense of the discontinuity of the solution map. To overcome the difficulties, we propose a new method based on the contradictory argument that reduces the problem to the analysis of the corresponding nonstationary Navier–Stokes equations and shows the existence of nonstationary solutions with strange large time behavior, if we suppose to contrary that the stationary problem is well-posed. PubDate: 2024-05-28 DOI: 10.1007/s40818-024-00174-z

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Abstract: Abstract In 2003, Klainerman and Nicolò [14] proved the stability of Minkowski in the case of the exterior of an outgoing null cone. Relying on the method used in [14], Caciotta and Nicolò [2] proved the stability of Kerr spacetime in external regions, i.e. outside an outgoing null cone far away from the Kerr event horizon. In this paper, we give a new proof of [2]. Compared to [2], we reduce the number of derivatives needed in the proof, simplify the treatment of the last slice, and provide a unified treatment of the decay of initial data which contains in particular the initial data considered by Klainerman and Szeftel in [20]. Also, concerning the treatment of curvature estimates, similar to [25], we replace the vectorfield method used in [2, 14] by \(r^p\) –weighted estimates introduced by Dafermos and Rodnianski in [8]. PubDate: 2024-05-07 DOI: 10.1007/s40818-024-00173-0

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Abstract: Abstract We study the asymptotics of complete Kähler-Einstein metrics on strictly pseudoconvex domains in \(\mathbb {C}^n\) and derive a convergence theorem for solutions to the corresponding Monge-Ampère equation. If only a portion of the boundary is analytic, the solutions satisfy Gevrey type estimates for tangential derivatives. A counterexample for the model linearized equation suggests that there is no local convergence theorem for the complex Monge-Ampère equation. PubDate: 2024-04-02 DOI: 10.1007/s40818-024-00171-2

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Abstract: Abstract The sphere is well-known as the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize the generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, we establish that generically the arrival time function of level set flow with spherical singularity has at most \(C^2\) regularity. PubDate: 2024-03-29 DOI: 10.1007/s40818-024-00170-3

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Abstract: Abstract We introduce the notions of static regular of type (I) and type (II) and show that they are sufficient conditions for local well-posedness of solving asymptotically flat, static vacuum metrics with prescribed Bartnik boundary data. We then show that hypersurfaces in a very general open and dense family of hypersurfaces are static regular of type (II). As applications, we confirm Bartnik’s static vacuum extension conjecture for a large class of Bartnik boundary data, including those that can be far from Euclidean and have large ADM masses, and give many new examples of static vacuum metrics with intriguing geometry. PubDate: 2024-03-05 DOI: 10.1007/s40818-024-00169-w

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Abstract: Abstract We study a free transmission problem driven by degenerate fully nonlinear operators. Our first result concerns the existence of a viscosity solution to the associated Dirichlet problem. By framing the equation in the context of viscosity inequalities, we prove regularity results for the constructed viscosity solution to the problem. Our findings include regularity in \( C^{1,\alpha }\) spaces, and an explicit characterization of \(\alpha \) in terms of the degeneracy rates. We argue by perturbation methods, relating our problem to a homogeneous, fully nonlinear uniformly elliptic equation. PubDate: 2024-02-20 DOI: 10.1007/s40818-024-00168-x

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Abstract: Abstract Given a measure \(\rho \) on a domain \(\Omega \subset {\mathbb {R}}^m\) , we study spacelike graphs over \(\Omega \) in Minkowski space with Lorentzian mean curvature \(\rho \) and Dirichlet boundary condition on \(\partial \Omega \) , which solve The graph function also represents the electric potential generated by a charge \(\rho \) in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer \(u_\rho \) of the associated action $$\begin{aligned} I_\rho (\psi ) \doteq \int _{\Omega } \Big ( 1 - \sqrt{1- D\psi ^2} \Big ) \textrm{d}x - \langle \rho , \psi \rangle \end{aligned}$$ among functions \(\psi \) satisfying \( D\psi \le 1\) , by the lack of smoothness of the Lagrangian density for \( D\psi = 1\) one cannot guarantee that \(u_\rho \) satisfies the Euler-Lagrange equation ( \(\mathcal{B}\mathcal{I}\) ). A chief difficulty comes from the possible presence of light segments in the graph of \(u_\rho \) . In this paper, we investigate the existence of a solution for general \(\rho \) . In particular, we give sufficient conditions to guarantee that \(u_\rho \) solves ( \(\mathcal{B}\mathcal{I}\) ) and enjoys \(\log \) -improved energy and \(W^{2,2}_\textrm{loc}\) estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of \(\rho \) to ensure the solvability of ( \(\mathcal{B}\mathcal{I}\) ). PubDate: 2024-01-29 DOI: 10.1007/s40818-023-00167-4

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Abstract: Abstract We prove global in time well-posedness for perturbations of the 2D stochastic Navier–Stokes equations $$\begin{aligned} \partial _t u + u \cdot \nabla u= & {} \Delta u - \nabla p + \zeta + \xi \;, \qquad u (0, \cdot ) = u_{0} \;,\\ {\text {div}}(u)= & {} 0 \;, \end{aligned}$$ driven by additive space-time white noise \( \xi \) , with perturbation \( \zeta \) in the Hölder–Besov space \(\mathcal {C}^{-2 + 3\kappa } \) , periodic boundary conditions and initial condition \( u_{0} \in \mathcal {C}^{-1 + \kappa } \) for any \( \kappa >0 \) . The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and tools from paracontrolled calculus. Our argument uses that the solution to the linear equation is a \( \log \) –correlated field, yielding a double exponential growth bound on the solution. Notably, our method does not rely on any explicit knowledge of the invariant measure to the SPDE, hence the perturbation \( \zeta \) is not restricted to the Cameron–Martin space of the noise, and the initial condition may be anticipative. Finally, we introduce a notion of weak solution that leads to well-posedness for all initial data \( u_{0}\) in \( L^{2} \) , the critical space of initial conditions. PubDate: 2023-12-27 DOI: 10.1007/s40818-023-00165-6

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Abstract: Abstract We prove asymptotic stability of the Poisson homogeneous equilibrium among solutions of the Vlasov–Poisson system in the Euclidean space \(\mathbb {R}^3\) . More precisely, we show that small, smooth, and localized perturbations of the Poisson equilibrium lead to global solutions of the Vlasov–Poisson system, which scatter to linear solutions at a polynomial rate as \(t\rightarrow \infty \) . The Euclidean problem we consider here differs significantly from the classical work on Landau damping in the periodic setting, in several ways. Most importantly, the linearized problem cannot satisfy a “Penrose condition”. As a result, our system contains resonances (small divisors) and the electric field is a superposition of an electrostatic component and a larger oscillatory component, both with polynomially decaying rates. PubDate: 2023-12-13 DOI: 10.1007/s40818-023-00161-w

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Abstract: Abstract In this paper, we address for the 2D Euler equations the existence of rigid time periodic solutions close to stationary radial vortices of type \(f_0( x )\textbf{1}_{{{\,\mathrm{\mathbb {D}}\,}}}(x)\) , with \({{\,\mathrm{\mathbb {D}}\,}}\) the unit disc and \(f_0\) being a strictly monotonic profile with constant sign. We distinguish two scenarios according to the sign of the profile: defocusing and focusing. In the first regime, we have scarcity of the bifurcating curves associated with lower symmetry. However in the focusing case we get a countable family of bifurcating solutions associated with large symmetry. The approach developed in this work is new and flexible, and the explicit expression of the radial profile is no longer required as in [41] with the quadratic shape. The alternative for that is a refined study of the associated spectral problem based on Sturm-Liouville differential equation with a variable potential that changes the sign depending on the shape of the profile and the location of the time period. Deep hidden structure on positive definiteness of some intermediate integral operators are also discovered and used in a crucial way. Notice that a special study will be performed for the linear problem associated with the first mode founded on Prüfer transformation and Kneser’s Theorem on the non-oscillation phenomenon. PubDate: 2023-12-12 DOI: 10.1007/s40818-023-00166-5

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Abstract: Abstract We prove a local existence theorem for the free boundary problem for a relativistic fluid in a fixed spacetime. Our proof involves an a priori estimate which only requires control of derivatives tangential to the boundary, which holds also in the Newtonian compressible case. PubDate: 2023-12-01 DOI: 10.1007/s40818-023-00164-7

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Abstract: Abstract We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, \( Q =M, \) Reissner–Nordström spacetime, as a solution to the Einstein–Maxwell equations. Our work uses and extends the framework [28, 32] of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon \( {\mathcal {H}}^+ \) . In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along \( {\mathcal {H}}^+ \) , the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component \( {\underline{\alpha }} \) not decaying asymptotically along the event horizon \( {\mathcal {H}}^+, \) a result previously unknown in the literature. PubDate: 2023-11-17 DOI: 10.1007/s40818-023-00158-5

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Abstract: Abstract The Obukhov–Corrsin theory of scalar turbulence [21, 54] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov’s K41 theory of fully developed turbulence [47]. The scaling analysis of Obukhov and Corrsin from 1949 to 1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field and an initial datum such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity within any fixed supercritical Obukhov-Corrsin regularity regime while also exhibiting anomalous dissipation. Our approach relies on a fine quantitative analysis of the interaction between the spatial scale of the solution and the scale of the Brownian motion which represents diffusion in a stochastic Lagrangian setting. This provides a direct Lagrangian approach to anomalous dissipation which is fundamental in order to get detailed insight on the behavior of the solution. Exploiting further this approach, we also show that for a velocity field in \(C^\alpha \) of space and time (for an arbitrary \(0 \le \alpha < 1\) ) neither vanishing diffusivity nor regularization by convolution provide a selection criterion for bounded solutions of the advection equation. This is motivated by the fundamental open problem of the selection of solutions of the Euler equations as vanishing-viscosity limit of solutions of the Navier-Stokes equations and provides a complete negative answer in the case of passive advection. PubDate: 2023-11-02 DOI: 10.1007/s40818-023-00162-9