Elliptic Estimates Research Articles

Global well-posedness of the three-dimensional free boundary problem for viscoelastic fluids without surface tension

In this paper, we consider the three-dimensional free boundary problem of incompressible and compressible neo-Hookean viscoelastic fluid equations in an infinite strip without surface tension, provided that the initial data is sufficiently close to the equilibrium state. By reformulating the problems in Lagrangian coordinates, we can get the stabilizing effect of elasticity. In both cases, we utilize the elliptic estimates to improve the estimates. Moreover, for the compressible case, we find there is an extra ODE structure that can improve the regularity of the free boundary, thus we can have the global well-posedness. To prove the global well-posedness for the incompressible case, we employ two-tier energy method introduced in [11][12][13] to compensate for the inferior structure.

Inverse problems of the wave equation for media with mixed but separated heterogeneous parts

In this article, the inverse problems for the wave equation in a medium in which multiple types of cavities and inclusion exist in a mixture are considered. From the point of view of the indicator function of the enclosure method, there are two types of heterogeneous parts: “minus group” and “plus group.” For example, cavities with the Dirichlet boundary condition belong to the minus group, while inclusions with smaller propagation velocity belong to the plus group. The heterogeneous part of the minus group gives a negative sign to the indicator function, and the heterogeneous part of the plus group gives a positive sign. In general, the presence of many types of heterogeneous parts causes cancelation of the sign of the indicator function. Such cases are referred to as “mixed cases.” Here, we consider the case that the shortest length obtained from the indicator function is attained only by heterogeneous parts of the same group. This case is called the “mixed but separated case,” and it is shown that the method of elliptic estimates developed by Ikehata works well. We also show that the case of a two‐layered background medium with a flat layer can be considered in the same way as the case of a homogeneous background medium.

Positive steady states in a two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity

– In this paper, we consider the following stationary two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity (0.1)−d1Δu=λu−u2−buv,x∈Ω,−div[d(w)∇v+χ(w)v∇w]=μv−v2−cuv,x∈Ω,−d3Δw=−κw+τu,x∈Ω,u=v=w=0,x∈∂Ωin a bounded smooth domain Ω⊂RN(N≥1), where λ,μ,b,c,d1,d3,κ,τ are positive constants, and (d(w),χ(w))∈[C1[0,∞)]2 with d(w),χ(w)>0 for all w≥0. Since there does not exist an immediate change variable that transforms (0.1) into a semilinear system when (0.1) is considered with d(w),α(w) being arbitrary functions in w, this makes the analysis of system (0.1) much more difficult. By the W2,p-estimate and the global bifurcation method, we obtain a bounded connected set of positive steady states joining the semitrivial solution of the form (u,0,w) with u,w>0 and the semitrivial solution of the form (0,v,0) with v>0, which provides the sufficient condition for the existence of positive steady states. Combining the eigenvalue theory, homogenization technique and various elliptic estimates, we also derive some sufficient conditions for the nonexistence of positive steady states.

Rough solutions of the relativistic Euler equations

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory.

Contact instantons with Legendrian boundary condition: A priori estimates, asymptotic convergence and index formula

In this paper, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential [Formula: see text]-convergence result at a puncture under the uniform [Formula: see text] bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. Leaving the study of [Formula: see text]-estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [27], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [27]–[29] and [36] containing applications to contact topology and contact Hamiltonian dynamics.

Hamilton and Li–Yau type gradient estimates for a weighted nonlinear parabolic equation under a super Perelman–Ricci flow

In this paper we derive elliptic and parabolic type gradient estimates for positive smooth solutions to a class of nonlinear parabolic equations on smooth metric measure spaces where the metric and potential are time dependent and evolve under a super Perelman–Ricci flow. A number of implications, notably, a parabolic Harnack inequality, a class of Hamilton type dimension-free gradient estimates and two general Liouville type theorems along with their consequences are discussed. Some examples and special cases are presented to illustrate the results.

Elliptic gradient estimate for the $p$−Laplace operator on the graph

Elliptic gradient estimate for the $p$−Laplace operator on the graph

Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds II: Integral curvature condition

We present a unified method for deriving differential Harnack inequalities for positive solutions to semilinear parabolic equations on compact manifolds and complete Riemannian manifolds, subject to an integral curvature condition. Specifically, we obtain the differential Harnack inequalities by solving a related system of ordinary differential equations. In addition to the case of scalar equations, we also establish an elliptic estimate for the heat flow under the same condition, which is a novel result for both harmonic map and heat equations. Many of the results presented here are nearly sharp, meaning they are sharp under the assumption of Ricci nonnegativity.

The seed-to-solution method for the Einstein constraints and the asymptotic localization problem

We establish the existence of a broad class of asymptotically Euclidean solutions to Einstein's constraint equations, whose asymptotic behavior at infinity is prescribed arbitrarily. The proposed seed-to-solution method encompasses vacuum as well as matter spaces, and relies on iterations based on the linearized Einstein operator and its dual. It generates a Riemannian manifold (with finitely many asymptotically Euclidean ends) from any seed data set consisting of(1) a Riemannian metric and a symmetric two-tensor and(2) a (density) field and a (momentum) vector field representing the matter content. We distinguish between tame or strongly tame seed data sets, depending whether the data provides a rough or an accurate asymptotic Ansatz at infinity. We encompass classes of metrics and matter fields with low decay (possibly with infinite ADM mass) or strong decay (with Schwarzschild behavior). Our analysis is motivated by Carlotto and Schoen's pioneering work on the localization problem for Einstein's vacuum equations. Dealing with metrics with low decay and, simultaneously, establishing estimates that include (and go beyond) harmonic decay require significantly new arguments which are developed in the present paper. We work in a weighted Lebesgue-Hölder framework adapted to the given seed data, and we analyze the nonlinear coupling between the Hamiltonian and momentum constraints. By establishing elliptic regularity estimates for the linearized Einstein operator and its dual, we uncover the novel notion of mass-momentum correctors which is related to the ADM mass-momentum of the manifold. We derive precise estimates for the difference between the seed data and the associate Einstein solution —a result that should be of interest for future numerical investigations. Furthermore, we introduce here and study the asymptotic localization problem in which we replace Carlotto-Schoen's exact localization requirement by an asymptotic condition at a super-harmonic rate. By applying our seed-to-solution method with a suitably constructed, parametrized family of seed data, we solve this problem by exhibiting mass-momentum correctors with harmonic decay.

Elliptic gradient estimates for a nonlinear equation with Dirichlet boundary condition

We prove local elliptic type gradient estimates for a nonlinear parabolic equation (including the fast diffusion equation and the porous medium equation) with Dirichlet boundary condition on Riemannian manifolds with compact boundary when the Ricci curvature and the mean curvature of boundary are bounded below. As an application, we obtain Liouville type results for certain ancient solutions to the nonlinear parabolic equation on Riemannian manifolds with compact boundary.

Deep learning‐based hybrid reconstruction algorithm for fibre instance segmentation from 3D x‐ray tomographic images

Abstract3D x‐ray tomography is a powerful scanning technique used for generating images of complex fibre structures. A novel machine‐learning algorithm to identify and separate individual fibres using 3D images is proposed in this article. The developed four‐step hybrid 3D fibre segmentation algorithm involves deep‐learning aided semantic segmentation that slices 3D images to create 2D images for fibre extraction, elliptical contour estimation combined with the marker‐controlled watershed algorithm for separating fibres from the background area, identifying individual fibres through 3D reconstruction, and, lastly, the 3D object refining approach based on outlier object detection and replacement. The proposed methodology is implemented on a real‐time sample of nylon fibre bundle under compression and its 3D x‐ray image volume to validate the performance. The results show its superior performance compared to off‐the‐shelf image processing algorithms in terms of precision, that is, with a validation accuracy greater than 90%, and efficiency, that is, preventing the need for a huge data set and reducing the complexity.

Local well-posedness of the free-boundary problem in compressible resistive magnetohydrodynamics

We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand-Shkoller to construct the approximation solutions. One of the key observations is that the Christodoulou-Lindblad type elliptic estimate together with magnetic diffusion not only gives the common control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to nonlinear parabolic equations involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman–Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry–Émery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed.

Late-time tails and mode coupling of linear waves on Kerr spacetimes

We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalar wave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angular frequencies ℓ≥1 and ℓ≥2. The ℓ-dependent asymptotics on Kerr spacetimes differ significantly from the non-rotating Schwarzschild setting (“Price's law”). The main differences with Schwarzschild are slower decay rates for higher angular frequencies and oscillations along the null generators of the event horizon. We introduce a physical space-based method that resolves the following two main difficulties for establishing ℓ-dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conserved charges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates in the full black hole exterior. This framework is suitable for addressing conflicting results on the numerology of Kerr late-time asymptotics appearing in the numerics literature and definitively determining the correct numerology.

Mode stability results for the Teukolsky equations on Kerr–anti-de Sitter spacetimes

We prove that there are no non-stationary (with respect to the Hawking vectorfield), real mode solutions to the Teukolsky equations on all -dimensional subextremal Kerr–anti-de Sitter spacetimes. We further prove that stationary solutions do not exist if the black hole parameters satisfy the Hawking–Reall bound and . We conclude with the statement of mode stability which preludes boundedness and decay estimates for general solutions which will be proven in a separate paper. Our boundary conditions are the standard ones which follow from fixing the conformal class of the metric at infinity and lead to a coupling of the two Teukolsky equations. The proof relies on combining the Teukolsky–Starobinsky identities with the coupled boundary conditions. In the stationary case the proof exploits elliptic estimates which fail if the Hawking–Reall bound is violated. This is consistent with the superradiant instabilities expected in that regime.

Analysis of contact Cauchy–Riemann maps III: Energy, bubbling and Fredholm theory

In [Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps I: A priori [Formula: see text] estimates and asymptotic convergence, Osaka J. Math. 55(4) (2018) 647–679; Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps II: Canonical neighborhoods and exponential convergence for the Morse-Bott case, Nagoya Math. J. 231 (2018) 128–223], the authors studied the nonlinear elliptic system [Formula: see text] without involving symplectization for each given contact triad [Formula: see text], and established the a priori [Formula: see text] elliptic estimates and proved the asymptotic (subsequence) convergence of the map [Formula: see text] for any solution, called a contact instanton, on [Formula: see text] under the hypothesis [Formula: see text] and [Formula: see text]. The asymptotic limit of a contact instanton is a ‘spiraling’ instanton along a ‘rotating’ Reeb orbit near each puncture on a punctured Riemann surface [Formula: see text]. Each limiting Reeb orbit carries a ‘charge’ arising from the integral of [Formula: see text]. In this paper, we further develop analysis of contact instantons, especially the [Formula: see text] estimate for [Formula: see text] (or the [Formula: see text]-estimate), which is essential for the study of compactification of the moduli space and the relevant Fredholm theory for contact instantons. In particular, we define a Hofer-type off-shell energy [Formula: see text] for any pair [Formula: see text] with a smooth map [Formula: see text] satisfying [Formula: see text], and develop the bubbling-off analysis and prove an [Formula: see text]-regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).

The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface $$\Sigma \simeq \overline{B_1} \subset {{\mathbb {R}}}^3$$ and the outgoing null hypersurface $${{\mathcal {H}}}$$ emanating from $${\partial }\Sigma $$ , we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in $$L^2$$ . The proof uses the bounded $$L^2$$ curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

A scale-critical trapped surface formation criterion for the Einstein-Maxwell system

In this study we show that, from arbitrarily dispersed initial data, both the focusing of electromagnetic fields and the concentration of gravitational waves could lead to the formation of trapped surfaces. We establish a scale–critical semi–global existence result from past null infinity for the Einstein–Maxwell system by assigning the signature for decay rates to both geometric quantities and Maxwell components. This result generalizes an approach of the first author in [4] on studying the Einstein vacuum equations to the physical Einstein-Maxwell system by employing additional elliptic estimates and geometric renormalizations and it also extends a result of Yu [18] and serves as the first scale-critical result for Einstein's equations coupled with matter.

Elliptic-type gradient estimates under integral Ricci curvature bounds

Let ( M n , g ) (\mathbf {M}^n, g) be an n n -dimensional complete Riemannian manifold. We prove that for any p > n 2 p>\frac {n}{2} , when k ( p , 1 ) k(p,1) is small enough, two certain elliptic-type gradient estimates hold for any positive solutions of the equation u t = Δ u + a u log ⁡ u \begin{equation*} u_{t}=\Delta u+au\log u \end{equation*} on geodesic balls B ( O , r ) B(O,r) in M n \mathbf {M}^n with 0 > r ≤ 1 0>r\leq 1 . Here the assumption that k ( p , 1 ) k(p,1) is small allows the situation where the manifold is collapsing. As applying, two forms of elliptic type gradient estimates to the heat equation on M n M^n under integral curvature conditions are obtained. Besides, each of the two forms of gradient estimate has its own advantages and differences (see Remark 4).

Local well-posedness for the motion of a compressible gravity water wave with vorticity

In this paper we prove the local well-posedness (LWP) for the 3D compressible Euler equations describing the motion of a liquid in an unbounded initial domain with moving boundary. The liquid is under the influence of gravity but without surface tension, and it is not assumed to be irrotational. We apply the tangential smoothing method introduced in Coutand-Shkoller [10,11] to construct the approximation system with energy estimates uniform in the smooth parameter. It should be emphasized that, when doing the nonlinear a priori estimates, we need neither the higher order wave equation of the pressure and delicate elliptic estimates, nor the higher regularity on the flow-map or initial vorticity. Instead, we adapt the Alinhac's good unknowns to the estimates of full spatial derivatives.