Well-posedness Theory Research Articles

Well-posedness for the surface quasi-geostrophic front equation

We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021 Pure Appl. Anal. 3 403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation’s nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.

Magneto-micropolar boundary layers theory in Sobolev spaces without monotonicity: well-posedness and convergence theory

Magneto-micropolar boundary layers theory in Sobolev spaces without monotonicity: well-posedness and convergence theory

Orbital instability of periodic waves for scalar viscous balance laws

The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital (nonlinear) instability in appropriate periodic Sobolev spaces. The analysis is based on the well-posedness theory, the smoothness of the data-solution map, and an abstract result of instability of equilibria under nonlinear iterations. The resulting instability criterion is applied to two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a local Hopf bifurcation around a critical value of the velocity. The second family comprises arbitrarily large period waves which arise from a homoclinic (global) bifurcation and tend to a limiting traveling pulse when their fundamental period tends to infinity. In the case of both families, the criterion is applied to conclude their orbital instability under the flow of the nonlinear viscous balance law in periodic Sobolev spaces with same period as the fundamental period of the wave.

Temperature dependent extensions of the Cahn–Hilliard equation

The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In this paper we focus on the dynamics of these binary media, when the underlying temperature is not constant.The aim of this paper is twofold. We first derive two distinct models that extend the classical Cahn-Hilliard equation with an evolutionary equation for the absolute temperature. Secondly, we analyse the local well-posedness of classical solution for one of these systems.Our modelling introduces the systems of PDEs by means of a general and unified formalism. This formalism couples standard principles of mechanics together with the main laws of thermodynamics. Our work highlights how certain assumptions on the transport of the temperature effect the overall physics of the systems.The variety of these thermodynamically consistent models opens the question of which one should be more appropriate. Our analysis shows that one of the derived models might be more desirable to the well-posedness theory of classical solutions, a property that might be natural as a selection criteria.We conclude our paper with an overview and comparison of our modelling formalism with some equations, which were previously derived in literature.

Spatial decay properties for a model in shear flows posed on the cylinder

We study spatial decay properties for solutions of the Pelinovski–Stepanyants equation posed on the cylinder. We establish the maximum polynomial decay admissible for solutions of such a model. It is verified that the equation on the cylinder propagates polynomial weights with different restrictions than the model set in R2. For example, a local well-posedness theory is deduced which contains the line solitary wave provided by solutions of the Benjamin–Ono equation extended to the cylinder. Key results for our analysis are obtained from a detailed study of the dispersive effects of the linear equation in weighted Sobolev spaces. The results in this paper appear to be one of the first studies of polynomial decay for nonlocal models in the cylinder.

Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations

We study the Cauchy problem for the inhomogeneous Hartree equation in this paper. Although its well-posedness theory has been extensively studied in recent years, much less is known compared to the classical Hartree model of homogeneous type. In particular, the problem of Sobolev initial data with the Sobolev critical index remains unsolved. The main contribution of this paper is to establish the local existence of solutions to the inhomogeneous equation in the critical cases. To do so, we obtain all possible Lp Strichartz estimates with singular weights.

The time-like minimal surface equation in Minkowski space: low regularity solutions

It has long been conjectured that for nonlinear wave equations that satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for the generic case. The aim of this article is to prove the first result in this direction, namely for the time-like minimal surface equation in the Minkowski space-time. Further, our improvement is substantial, namely by 3/8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$3/8$\\end{document} derivatives in two space dimensions and by 1/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1/4$\\end{document} derivatives in higher dimensions.

The Euler–Poisswell/Darwin equation and the asymptotic hierarchy of the Euler–Maxwell equation

In this paper we introduce the (unipolar) pressureless Euler–Poisswell equation for electrons as the O ( 1 / c ) semi-relativistic approximation and the (unipolar) pressureless Euler–Darwin equation as the O ( 1 / c 2 ) semi-relativistic approximation of the (unipolar) pressureless Euler–Maxwell equation. In the “infinity-ion-mass” limit, the unipolar Euler–Maxwell equation arises from the bipolar Euler–Maxwell equation, describing a coupled system for a plasma of electrons and ions. The non-relativistic limit c → ∞ of the Euler–Maxwell equation is the repulsive Euler–Poisson equation with electric force. We propose that the Euler–Poisswell equation, where the Euler equation with electric force is coupled to the magnetostatic O ( 1 / c ) approximation of Maxwell’s equations, is the correct semi-relativistic O ( 1 / c ) approximation of the Euler–Maxwell equation. In the Euler–Poisswell equation the fluid dynamics are decoupled from the magnetic field since the Lorentz force reduces to the electric force. The first non-trivial interaction with the magnetic field is at the O ( 1 / c 2 ) level in the Euler–Darwin equation. This is a consistent and self-consistent model of order O ( 1 / c 2 ) and includes the full Lorentz force, which is relativistic at O ( 1 / c 2 ). The Euler–Poisswell equation also arises as the semiclassical limit of the quantum Pauli–Poisswell equation, which is the O ( 1 / c ) approximation of the relativistic Dirac–Maxwell equation. We also present a local wellposedness theory for the Euler–Poisswell equation. The Euler–Maxwell system couples the non-relativistic Euler equation and the relativistic Maxwell equations and thus it is not fully consistent in 1 / c. The consistent fully relativistic model is the relativistic Euler–Maxwell equation where Maxwell’s equations are coupled to the relativistic Euler equation – a model that is neglected in the mathematics community. We also present the relativistic Euler–Darwin equation resulting as a O ( 1 / c 2 ) approximation of this model.

Global well-posedness of the 1d compressible Navier–Stokes system with rough data

In this paper, we study the global well-posedness problem for the 1d compressible Navier–Stokes systems (cNSE) in gas dynamics with rough initial data. First, Liu and Yu (2022) [30] established the global well-posedness theory for the 1d isentropic cNSE with initial velocity data in BV space. Then, it was extended to the 1d full cNSE with initial velocity and temperature data in BV space by Wang et al. (2022) [31]. We improve the global well-posedness result of Liu and Yu with initial velocity data in W2γ,1 space; and of Wang–Yu–Zhang with initial velocity data in L2∩W2γ,1 space and initial data of temperature in W˙−23,65∩W˙2γ−1,1 for any γ>0arbitrarily small. Our essential ideas are based on establishing various “end-point” smoothing estimates for the 1d parabolic equation.

Stochastic fractional conservation laws

In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws with stochastic forcing. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.

Uniqueness and uniform structural stability of Poiseuille flows in a two-dimensional strip

In this paper, we prove the uniform nonlinear structural stability of Poiseuille flows with arbitrarily large flux for the steady Navier-Stokes system in a two-dimensional periodic strip when the period is not very large. The key point is to establish the a priori estimate for the corresponding linearized problem via the careful analysis for the associated boundary layers. Furthermore, the well-posedness theory for the Navier-Stokes system is also proved even when the L2-norm of the external force is large. These results prove the uniqueness of solutions for the steady Navier-Stokes system even when the flux is large and the flow is not symmetric. In particular, if the vertical velocity is suitably small where the smallness is independent of the flux, then Poiseuille flow is the unique solution of the steady Navier-Stokes system in the periodic strip.

Local-in-time well-posedness theory for the inhomogeneous MHD boundary layer equations without resistivity in lower regular Sobolev space

In this paper, we consider the well-posedness theory in lower regular Sobolev space for the Prandtl-type equation arising from the inhomogeneous incompressible MHD equation in dimension two. Due to the degeneracy of horizontal dissipation, the well-posedness result of MHD boundary layer equation is established in higher regular Sobolev space rather than lower one generally. By using the anisotropic Sobolev inequality and characteristic line method, the well-posedness result of inhomogenous MHD Prandtl-type equation can be established under the H2−framework.

The global classical solution to compressible system with fractional viscous term

We consider the initial value problem to the fractional system of motions for compressible viscous fluids in this paper. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the large-time behavior of the solution, where solutions converge to a constant steady state exponentially in time.

Optimal control problems in transport dynamics with additive noise

Motivated by the applications in leader-follower multi-agent dynamics, a class of optimal control problems is investigated, where the goal is to influence the behavior of a given population through another controlled one interacting with the first. The evolution of the systems obeys a non-linear Fokker–Planck-type equation as diffusive terms accounting for randomness in the evolution are taken into account. A well-posedness theory under very low regularity of the control vector fields is developed, as well as a rigorous mean-field limit of a stochastic particle approximation.

Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift

Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate parabolic-hyperbolic PDE with irregular drift. We develop a robust well-posedness theory for such PDEs in energy-critical spaces based on concepts of renormalized solutions and the equation’s kinetic form. We establish these properties by proving that renormalized solutions are equivalent to classical weak solutions, extending concepts of (DiPerna and Lions in Invent. Math. 98(3):511–547, 1989; Ambrosio in Invent. Math. 158(2):227–260, 2004) to the nonlinear setting.The relevance of the results toward large deviations in interacting particle systems is demonstrated by applications to the identification of l.s.c. envelopes of restricted rate functions, to zero noise large deviations for conservative SPDE, and to the Gamma -convergence of rate functions. The first of these solves a long-standing open problem in the large deviations for zero range processes. The second makes rigorous an informal link between the non-equilibrium statistical mechanics approaches of macroscopic fluctuation theory and fluctuating hydrodynamics.

On a Kinetic Equation Describing the Behavior of a Gas Interacting Mainly with Radiation

In this article we study a kinetic model which describes the interaction between a gas and radiation. Specifically, we consider a scaling limit in which the interaction between the gas and the photons takes place much faster than the collisions between the gas molecules themselves. We prove in the homogeneous case that the solutions of the limit problem solve a kinetic equation for which a well-posedness theory is considered. The proof of the convergence to a new kinetic equation is obtained analyzing the dynamics of the gas–photon system near the slow manifold of steady states.

Global Well-Posedness of Shock Front Solutions to One-Dimensional Piston Problem for Combustion Euler Flows

This paper is devoted to the well-posedness theory of piston problem for compressible combustion Euler flows with physical ignition condition. A significant combustion phenomenon called detonation will occur provided that the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In the present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure and from the characteristic boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary, and exothermic reaction and then obtain the -stability of combustion wave solutions. Our results reveal that one-dimensional Zeldovich–von Neumann–Döring detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.

Propagation of Moments for Large Data and Semiclassical Limit to the Relativistic Vlasov Equation

We investigate the semiclassical limit from the semirelativistic Hartree–Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form , , , and , with the convention if . For mixed states, we show convergence in the Schatten norms with an explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [E. Dietler, S. Rademacher, and B. Schlein, J. Stat. Phys., 172 (2018), pp. 398–433], where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions.

Propagation of Rough Initial Data for Navier–Stokes Equation

The purpose of this paper is to study the quantitative structure of a weak solution for an initial value problem of the compressible Navier–Stokes equation in the class of BV functions. In [T.-P. Liu and S.-H. Yu, Comm. Pure Appl. Math., 75 (2022), pp. 223–348] the precise pointwise structures of Green’s functions for various linearized problems are established, which serve as basic tools to establish the global existence of weak solutions to the Navier–Stokes equations with initial data as a small perturbation of a constant state in the BV class. In this paper, we obtain the time asymptotic pointwise structure of the weak solutions. It thus completes the initial motivation for the earlier paper on viewing the solution behavior as part of the well-posedness theory.

Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker–Smale dynamics. Our approach is inspired by the work of Brenier and Grenier on the pressureless Euler equations.