General Initial Data Research Articles

Global solutions to the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner system with a class of large initial data

In this paper, we consider the global well-posedness of the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner (PTT) system with general initial data in the critical Besov spaces. The question of whether or not PTT system is globally well posed for large data is still open. When the density ρ is away from zero, we denote by ϱ:=1ρ−1. More precisely, we prove that the PTT system admits a unique global solution, provided that the initial data ϱ 0, the initial horizontal velocity uh0 , the product ωu30 of the coupling parameter ω and the initial vertical velocity u 30, and the initial symmetric tensor of constraints τ 0 are sufficiently small. In particular, this result includes the global well-posedness of the PTT system for small initial data in the case where ω∈[0,1) and for large initial vertical velocity in the case where ω tends to zero. As a by-product, our results can be applied to the so-called Oldroyd-B system.

Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers.

Competing effects in fourth‐order aggregation–diffusion equations

AbstractWe give sharp conditions for global in time existence of gradient flow solutions to a Cahn–Hilliard‐type equation, with backwards second‐order degenerate diffusion, in any dimension and for general initial data. Our equation is the 2‐Wasserstein gradient flow of a free energy with two competing effects: the Dirichlet energy and the power‐law internal energy. Homogeneity of the functionals reveals critical regimes that we analyse. Sharp conditions for global in time solutions, constructed via the minimising movement scheme, also known as JKO scheme, are obtained. Furthermore, we study a system of two Cahn–Hilliard‐type equations exhibiting an analogous gradient flow structure.

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

We consider the magnetic nonlinear inhomogeneous Schrödinger equationi∂tu−(−i∇+α|x|2(−x2,x1))2u=±|x|−ϱ|u|p−1u,(t,x)∈R×R2, where α∈R∖Z,ϱ>0,p>1. We establish a dichotomy between global existence and scattering versus finite-time blow-up for energy solutions below the ground state threshold in the inter-critical regime. The scattering result is obtained by employing the novel approach developed by Dodson and Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Am. Math. Soc. (2017)). We employ Tao's scattering criteria and Morawetz estimates as the foundation of this method. The novelty of our approach lies in two aspects: firstly, we explore the case of nonzero ϱα, and secondly, we consider general energy initial data that need not be radially symmetric. The specific instance of α=0, known as INLS, has garnered significant attention in recent years. Additionally, X. Gao and C. Xu recently investigated the case of ϱ=0 in the homogeneous regime, establishing scattering theory for spherically symmetric data in their work (Scattering theory for NLS with inverse-square potential in 2D, J. Math. Anal. Appl. (2020)). In the radial framework, the aforementioned problem can be translated to the INLS with an inverse square potential, which has been extensively studied in higher-dimensional spaces. The Hardy inequality ‖|x|−1f‖L2(RN)≤2N−2‖∇f‖L2(RN), which establishes the norm equivalence ‖f‖H1≃‖f‖H1+‖|x|−1f‖L2, does not hold in two space dimensions. Consequently, it remains unclear how to handle the NLS with an inverse square potential in the H1 framework for two-dimensional space. This article stands out as the first one to address the NLS with Aharonov-Bohm magnetic potential within the inhomogeneous regime, specifically when ϱ≠0.

Horizon phase spaces in general relativity

We derive a prescription for the phase space of general relativity on two intersecting null surfaces using the null initial value formulation. The phase space allows generic smooth initial data, and the corresponding boundary symmetry group is the semidirect product of the group of arbitrary diffeomorphisms of each null boundary which coincide at the corner, with a group of reparameterizations of the null generators. The phase space can be consistently extended by acting with half-sided boosts that generate Weyl shocks along the initial data surfaces. The extended phase space includes the relative boost angle between the null surfaces as part of the initial data.We then apply the Wald-Zoupas framework to compute gravitational charges and fluxes associated with the boundary symmetries. The non-uniqueness in the charges can be reduced to two free parameters by imposing covariance and invariance under rescalings of the null normals. We show that the Wald-Zoupas stationarity criterion cannot be used to eliminate the non-uniqueness. The different choices of parameters correspond to different choices of polarization on the phase space. We also derive the symmetry groups and charges for two subspaces of the phase space, the first obtained by fixing the direction of the normal vectors, and the second by fixing the direction and normalization of the normal vectors. The second symmetry group consists of Carrollian diffeomorphisms on the two boundaries.Finally we specialize to future event horizons by imposing the condition that the area element be non-decreasing and become constant at late times. For perturbations about stationary backgrounds we determine the independent dynamical degrees of freedom by solving the constraint equations along the horizons. We mod out by the degeneracy directions of the presymplectic form, and apply a similar procedure for weak non-degeneracies, to obtain the horizon edge modes and the Poisson structure. We show that the area operator of the black hole generates a shift in the relative boost angle under the Poisson bracket.

Nonequilibrium–diffusion limit of the compressible Euler radiation model in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$

This article studies the nonequilibrium–diffusion limit of the compressible Euler model arising in radiation hydrodynamics in with the general initial data. Combining the moment method with Hilbert expansion, we show that the radiative intensity can be approximated by the sum of interior solution and initial layer. We also show that the solution satisfied by the density, temperature, and velocity can be approximated by the interior solutions. Our results can be considered as an extension from in arXiv.2312.15208 by Ju, Li, and Zhang to . In contrast to arXiv.2312.15208, we get the exact convergence rates by studying the error system derived from the primitive system, the zeroth‐order to the second‐order about the radiative intensity, and the zeroth‐order about the hydrodynamics.

The QG limit of magnetohydrodynamic rotating shallow water system

Magnetohydrodynamic rotating shallow water system (MRSW) is a proposed model for a thin layer of electrically conducting fluid, which plays an important role in astrophysical plasma studies. For the spatial periodic domain, a mathematically rigorous framework is developed for deriving reduced systems for MRSW equations with general unbalanced initial data. It is shown that the reduced slow dynamics are the magnetohydrodynamic quasi-geostrophic equations.

Exact global control of small divisors in rational normal form *

Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schrödinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.

Mean curvature flow with generic low-entropy initial data

Mean curvature flow with generic low-entropy initial data

Mean curvature flow with generic initial data

We show that the mean curvature flow of generic closed surfaces in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{3}$\\end{document} avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{4}$\\end{document} is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

Fast rotation and inviscid limits for the SQG equation with general ill-prepared initial data

Fast rotation and inviscid limits for the SQG equation with general ill-prepared initial data

Long time stability for the derivative nonlinear Schrödinger equation

In this paper, we consider the long time dynamics of the solutions of the derivative nonlinear Schrödinger equation on one dimensional torus without external parameters. By using rational normal form, we prove the long time stability for generic small initial data.

On the hierarchy and fine structure of blowups and gradient catastrophes for multidimensional homogeneous Euler equation *

Blowups of derivatives and gradient catastrophes for the n-dimensional homogeneous Euler equation are discussed. It is shown that, in the case of generic initial data, the blowups exhibit a fine structure in accordance with the admissible ranks of certain matrix generated by the initial data. Blowups form a hierarchy composed by n + 1 levels with the singularity of derivatives given by ∂ui/∂xk∼|δx|−(m+1)/(m+2) , m=1,…,n along certain critical directions. It is demonstrated that in the multi-dimensional case there is certain bounded linear superposition of blowup derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis.

Existence and uniqueness of the global conservative weak solutions for a cubic Camassa-Holm type equation

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v=2arctan⁡ux along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u0∈H1(R) was proved.

Global existence of dissipative solutions to the Camassa–Holm equation with transport noise

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.

Diffusion asymptotics of a coupled model for radiative transfer: general initial data

Diffusion asymptotics of a coupled model for radiative transfer: general initial data

On the regularity of the De Gregorio model for the 3D Euler equations

We study the regularity of the De Gregorio (DG) model \omega_{t} + u\omega_{x} = u_{x} \omega on S^{1} for initial data \omega_{0} with period \pi and in class X : \omega_{0} is odd and \omega_{0} \leq 0 (or \omega_{0} \geq 0 ) on [0,\pi/2] . These sign and symmetry properties are the same as those of the smooth initial data that lead to singularity formation of the De Gregorio model on \mathbb{R} or the generalized Constantin–Lax–Majda (gCLM) model on \mathbb{R} or S^{1} with a positive parameter. Thus, to establish global regularity of the DG model for general smooth initial data, which is a conjecture on the DG model, an important step is to rule out potential finite time blowup from smooth initial data in X . We accomplish this by establishing a one-point blowup criterion and proving global well-posedness for initial data \omega_{0} \in H^{1} \cap X with \omega_{0}(x) x^{-1}\in L^{\inf} . On the other hand, for any \alpha \in (0,1) , we construct a finite time blowup solution from a class of initial data with \omega_{0} \in C^{\alpha}\cap C^{\infty}(S^{1} \setminus \{0\}) \cap X . Our results imply that singularities developed in the DG model and the gCLM model on S^{1} can be prevented by stronger advection.

Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordström Spacetime

We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, |Q|=M,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ |Q|=M, $$\\end{document} 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 H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}. In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along H+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+ $$\\end{document}, 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 α̲\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\underline{\\alpha }} $$\\end{document} not decaying asymptotically along the event horizon H+,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {H}}^+, $$\\end{document} a result previously unknown in the literature.

Numerical study of fractional Camassa–Holm equations

A numerical study of fractional Camassa–Holm equations is presented. Smooth solitary waves are constructed numerically. Their stability is studied as well as the long time behavior of solutions for general localized initial data from the Schwartz class of rapidly decreasing functions. The appearance of dispersive shock waves is explored.

On the global well‐posedness and striated regularity of the 2D Boussinesq‐MHD system

In this paper, we investigate the global existence and uniqueness of strong solutions to the two‐dimensional Boussinesq‐MHD equations without the temperature diffusion for general initial data in with . As an application, we shall prove the propagation of the regularity of the interface between fluids with different temperatures.