Local Well-posedness Research Articles

An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions

Abstract Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green’s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

Blow-up issues and peakons for a two-component high-order Fokas–Olver–Rosenau–Qiao system

In this paper, we investigate the blow-up issues and peakons for a two-component high-order Fokas–Olver–Rosenau–Qiao system, which can be view as a multi-component extension of the high-order Fokas–Olver–Rosenau–Qiao equation. We first address the local well-posedness of the Cauchy problem in the framework of the Sobolev and Besov spaces. Next, we derive the precise blow-up mechanism for strong solutions through transport equation theory, utilizing the system’s fine structure and conservation law, we present two new blow-up results about certain initial profiles in finite time. Finally, peakon solutions are discussed as well. The results are helpful to understand how two-component and high-order nonlinearities affect the blow-up solutions.

Global well-posedness, ill-posedness and long time behavior of solutions of the surface electromigration equation

Abstract This paper is concerned with the Cauchy problem and the long time behavior of the two-dimensional surface electromigration (SEM) equation. Firstly, we prove that the SEM equation is locally well-posed in H s ( R 2 ) with s > − 1 4 , and globally well-posed in L 2 ( R 2 ) without smallness assumption. Secondly, we prove that the SEM equation is ill-posed in H s ( R 2 ) with s < − 1 4 in the sense that the solution map is not C 2. Finally, we get a local energy decay result of the global solution in a growing unbounded in time region but not containing the soliton region. Our well-posedness results extend the latest findings in Linares et al (2021 Nonlinearity 34 5213–33), where the authors proved that the SEM equation is locally well-posed in H s ( R 2 ) with s > 1 2 . The key ingredient in this paper is to rewrite the linear and nonlinear part in the SEM equation into symmetric separate dispersive terms on the two space variables and some divergence form, respectively. After this invertible process, we use two bilinear estimates in the modified Fourier restriction space introduced in Kinoshita (2021 Ann. Inst. Henri Poincare C 38 451–505) to relax the local well-posedness index from s > 1 2 to s > − 1 4 .

Local Well-posedness for the Kinetic MMT Model

Local Well-posedness for the Kinetic MMT Model

Well-posedness of Keller–Segel systems on compact metric graphs

Chemotaxis phenomena govern the directed movement of microorganisms in response to chemical stimuli. In this paper, we investigate two Keller–Segel systems of reaction–advection–diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic-type Keller–Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions.

A Dynamical Yukawa2 Model

We prove local (in space and time) well-posedness for a mildly regularised version of the stochastic quantisation of the Yukawa2 Euclidean field theory with a self-interacting boson. Our regularised dynamic is still singular but avoids non-local divergences, allowing us to use a version of the Da Prato–Debussche argument (Da Prato and Debussche in Ann Probab 31(4):1900–1916, 2003. https://doi.org/10.1214/aop/1068646370). This model is a test case for a non-commutative probability framework for formulating the kind of singular SPDEs arising in the stochastic quantisation of field theories mixing both bosons and fermions.

Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening

Abstract The Maxwell-Bloch system of equations with inhomogeneous broadening is studied, and the local and global well-posedness of the corresponding initial-boundary value problem is established by taking advantage of the integrability of the system and making use of the corresponding inverse scattering transform (IST). A key ingredient in the analysis is the L 2 {L}^{2} -Sobolev bijectivity of the direct and IST established by Xin Zhou for the focusing Zakharov-Shabat problem.

Self-organized hydrodynamic model with density-dependent velocity: local well-posedness and the limit from self-organized kinetic model

Self-organized hydrodynamic model with density-dependent velocity: local well-posedness and the limit from self-organized kinetic model

Radial solutions of initial boundary value problems of nonlinear Schrödinger equations in ℝn

The article studies an initial boundary valueproblem (ibvp) for the radial solutions of the nonlinear Schrödinger (NLS) equation in a radially symmetric region $\Omega\in \mathbb R^n$ with boundaries. All such regions can be classified into three types: a ball Ω0 centred at origin, a region Ω1 outside a ball, and an n-dimensional annulus Ω2. To study the well-posedness of those ibvps, the function spaces for the boundary data must be specified in terms of the solutions in appropriate Sobolev spaces. It is shown that when $\Omega = \Omega_1$ , the ibvp for the NLS equation is locally well-posed in $ C( [0, T^*]; H^s(\Omega_1))$ if the initial data is in $H^s(\Omega_1)$ and boundary data is in $ H^{\frac{2s+1}{4}}(0, T)$ with $s \geq 0$ . This is the optimal regularity for the boundary data and cannot be improved. When $\Omega = \Omega_2$ , the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_2))$ if the initial data is in $ H^s(\Omega_2)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with $s \geq 0$ . In this case, the boundary data requires $1/4$ more derivative compared to the case when $\Omega = \Omega_1$ . When $\Omega = \Omega_0$ with n = 2 (the case with n > 2 can be discussed similarly), the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_0))$ if the initial data is in $ H^s(\Omega_0)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with s > 1 (or $s \gt n/2$ ). Due to the lack of Strichartz estimates for the corresponding boundary integral operator with $ 0 \leq s \leq 1$ , the local well-posedness can only be achieved for s > 1. It is noted that the well-posedness results on Ω0 and Ω2 are the first ones for the ibvp of NLS equations in bounded regions of higher dimension.

Global Well-Posedness for Supercritical SQG With Perturbations of Radially Symmetric Data

Abstract We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial data remains open. Motivated by enhanced decay in radial solutions, we aim to establish global well-posedness for small perturbations of potentially large radial data. Our main result shows that for small perturbations of radial data, the SQG equation admits a unique global solution.

The local well-posedness for the dispersion generalized Camassa–Holm equation

In this paper, we establish the local well-posedness of the Cauchy problem for a dispersion generalized Camassa–Holm equation. The present generalization is obtained by replacing the operator ( 1 − ∂ x 2 ) of the Camassa–Holm equation with ( 1 − L ∂ x 2 ) where L is a positive differential operator with order p, an even positive integer. We follow Kato's semigroup approach for quasi-linear evolution equations but use L-dependent operators and norms. We show that the Cauchy problem is locally well-posed in a Banach space for which the norms are equivalent to Sobolev space norms and the regularity index has a threshold depending on p. Considering the special cases of the operator L, we verify that our results are consistent with those presented in the literature.

Fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity: Well-posedness, blow up and asymptotic stability

Considered herein is the initial–boundary value problem for a fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity given by ut+(−Δ)su+(−Δ)sut=∫0tg(t−τ)(−Δ)su(τ)dτ+uln|u| under different initial energy levels. The local well-posedness of weak solution is firstly established by using Galerkin approximation and contraction mapping principle at arbitrary initial energy level. Secondly, the global well-posedness, polynomial and exponential energy decay estimates, finite time blow up are investigated at low initial energy level (u0∈Wδ,1) by utilizing modified potential well theory, Galerkin approximation, perturbed energy method, differential–integral inequality technique etc. Subsequently, based on the above conclusions of low initial energy, the global existence, polynomial and exponential energy decay estimates and finite time blow up are also derived at critical initial energy level (u0∈Wδ,2) by introducing some new approximation methods and techniques. Here, the sets Wδ,i(i=1,2) defined in Section 2.2 denote potential well families involving the parameter δ>0. Finally, we give a series of numerical examples used to illuminate above theoretical results.

Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation

Abstract This article presents the application of the Faedo–Galerkin compactness method to establish the local well-posedness of the Newell–Whitehead–Segel equation. By analyzing a finite-dimensional approximate problem, the existence and uniqueness of a local solution were demonstrated. A priori estimates were derived, enabling the transition to the limit and the recovery of the original problem’s local solution. The study further proves the uniqueness and continuous dependence of the solution on initial data. Additionally, under certain conditions, it is shown that the energy norm of the solution decays exponentially over time, and the L 2 {L}^{2} -norm of the time derivative of the solution approaches zero asymptotically.

A uniqueness theory on determining the nonlinear energy potential in phase-field system

Abstract The phase-field system is a nonlinear model that has significant applications in material sciences. In this paper, we are concerned with the uniqueness of determining the nonlinear energy potential in a phase-field system consisting of Cahn–Hilliard and Allen–Cahn equations. This system finds widespread applications in the development of alloys engineered to withstand extreme temperatures and pressures. The goal is to reconstruct the nonlinear energy potential through the measurements of concentration fields. We establish the local well-posedness of the phase-field system based on the implicit function theorem in Banach spaces. Both of the uniqueness results for recovering time-independent and time-dependent energy potential functions are provided through the higher order linearization technique.

Strongly hyperbolic quasilinear systems revisited, with applications to relativistic fluid dynamics

We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest.

Two-dimensional gravity waves at low regularity I: Energy estimates

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and develop the techniques to prove a new class of energy estimates, which we call balanced cubic estimates . This yields a key improvement over the earlier cubic estimates of Hunter–Ifrim–Tataru (2016), while preserving their scale-invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using any Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness, drastically improving earlier results obtained by Alazard–Burq–Zuily (2014, 2018), Hunter–Ifrim–Tataru (2016) and Ai (2019).

Well-posedness of the Cauchy problem for the fourth-order nonlinear Schrödinger equation

The sharp local well-posedness for the one dimensional fourth-order nonlinear Schrödinger equation is established in the Sobolev space Hs(R) for s≥12, which improves the results in Huo and Jia (2007). In addition, we prove that this equation cannot be solved by an iteration scheme based on the Duhamel formula in Hs(R) for s<12. Our method relies upon the Bourgain space and a crucial bilinear estimate, which avoids the tedious classification of the location to the highest dispersion modulation.

Remark on the Local Well-Posedness of Compressible Non-Newtonian Fluids with Initial Vacuum

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in https://doi.org/10.1007/s00208-021-02301-8 can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in https://doi.org/10.1016/j.matpur.2003.11.004 for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic W2,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,p}$$\\end{document}-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

The Local Well-Posedness of the Relativistic Vlasov–Maxwell–Landau System with the Specular Reflection Boundary Condition

The Local Well-Posedness of the Relativistic Vlasov–Maxwell–Landau System with the Specular Reflection Boundary Condition

Dependence on initial data for a stochastic modified two-component Camassa-Holm system

We study a stochastic modified two-component Camassa-Holm equation on R. We establish a local well-posedness result in the sense of Hadamard, i.e. existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Motivated by the work of Miao et al. (2024) [29], we show that the solution map y0↦y(t) defined by the corresponding Cauchy problem is weakly unstable, due to either a lack of strong stability in the exiting time or the absence of uniformly continuous dependence on the initial data.