Small Initial Data Research Articles

Global classical solutions to equatorial shallow-water equations

In this article, we study the equatorial shallow-water equations with slip boundary condition in bounded domain. By exploring the dissipative structures of the system, we obtaining a priori estimates of the solution for small initial data. Then the existence of classical global solutions and exponential stability results are given.
 For more inofrmation see https://ejde.math.txstate.edu/Volumes/2023/62/abstr.html

Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B˙p,12p(R2)×B˙p,q2p−1(R2). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B˙p,12p(R2)×B˙p,12p−1(R2) for p<4 and ill-posed in B˙p,12p(R2)×B˙p,12p−1(R2) for p>4. In this paper, we prove that this system is ill-posed for the critical case p=4 in the sense of “norm inflation”. Furthermore, we also show that the system is ill-posed in B˙4,112(R2)×B˙4,q−12(R2) for any q≠2.

Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space : \begin{align} \left\{\begin{array}{ll} \partial_{t}u=\Delta_{\mathbb{H}^{n}} u+ f(u, t) & \hbox{ in }~ \mathbb{H}^{n}\times (0, T), \\ \quad u =u_{0} & \hbox{ in }~ \mathbb{H}^{n}\times \{0\}. \end{array}\right. \tag{0.1} \end{align} We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, (0.1) exhibits Fujita phenomena for the exponential weight $h(t) = e^{\mu t},$ i.e., there exists a critical exponent $ \mu^*$ such that if $\mu > \mu^*$, then all non-negative solutions blow-up in finite time and if $\mu \leq \mu^*$, there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that (0.1) with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.

Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems

Abstract We study the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction. In both L 1 {L^{1}} and L ∞ {L^{\infty}} frameworks, we will first show the global existence of classical solutions with small initial data, and then prove that the global solution will scatter to free linear waves and study the rigidity aspect of the scattering problem. We also show the inverse scattering result: The scattering field can determine the global solution uniquely, in the L 1 {L^{1}} 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.

Stability of the 2D anisotropic micropolar Rayleigh–Bénard problem with one direction dissipation

In this paper, we investigate the initial value problem for the 2D anisotropic micropolar Rayleigh–Bénard problem with one direction dissipation. But there are only vertical kinematic viscosity, horizontal angular viscosity, and diffusivity; some nonlinear terms cannot be controlled. In addition to the classical energy functional, we introduce the energy functional to help us dominate the nonlinear parts and make the energy closed. Then based on the energy method and the bootstrapping argument, global smooth solutions are proved under the assumptions of small initial data.

Global Classical Solutions and the Classical Limit of the Non-Relativistic Vlasov-Darwin System with Small Initial Data

Global Classical Solutions and the Classical Limit of the Non-Relativistic Vlasov-Darwin System with Small Initial Data

Global strong solutions and large time behavior to a micro-macro model for compressible polymeric fluids near equilibrium

In this paper, we mainly study the global strong solutions to a micro-macro model for compressible polymeric fluids with small initial data and their long-time decay rates of all order spatial derivatives. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first present that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for d≥2. Moreover, for d≥3, we show a new critical Fourier estimation that allows us to derive the long-time decay rates of the L2 norm for all order spatial derivatives.

Small Knudsen rate of convergence to contact wave for the Landau equation

In this paper, we study the hydrodynamic limit to contact discontinuity of the compressible Euler system for the Landau equation with the physical Coulomb interaction as the Knudsen number ε>0 is vanishing. Our results are twofold. First, we construct the unique global-in-time solution to the Landau equation with suitably small initial data as ε>0 is sufficiently small. Second, we prove that the solution of the Landau equation converges to a local Maxwellian whose fluid quantities are the contact discontinuity of the corresponding Euler system uniformly away from the discontinuity as ε→0. Moreover, the uniform convergence rate is also obtained. The main idea is to introduce a new time-velocity weigh function to induce an extra quartic dissipation term for treating the large-velocity growth in the nonlinear estimates due to degeneration of the linearized Landau operator in the Coulomb case.

Global strong solution of 3D inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity

In this paper, we investigate the Cauchy problem of the three-dimensional inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity. For the case μ(ρ)=ρ, we prove the global existence of strong solution for small initial data satisfying the compatibility condition. Furthermore, we also obtain the large time algebraic decay rates of the velocity. The innovation of this article is that it does not require a positive lower bound for viscosity.

Long‐time dynamics of Ericksen–Leslie system on 𝕊2

In this paper, we study the long‐time behavior of full Ericksen–Leslie system modeling the hydrodynamics of nematic liquid crystals between two dimensional unit spheres. Under a weaker assumption for Leslie's coefficients, we give the key energy inequality for the global weak solution. At last, inspired by the conditions on the simplified system, we establish several sufficient conditions which guarantee the uniform convergence of the system in and spaces as time tends to infinity under small initial data.

Convergence analysis from the indirect signal production to the direct one

This paper aims to provide the rigorous convergence analysis of the chemotaxis system{∂nϵ∂t=Δnϵ−∇⋅(nϵ∇cϵ),τ1∂cϵ∂t=Δcϵ−cϵ+wϵ,τ2∂wϵ∂t=−wϵ+nϵ in a smoothly bounded planar domain Ω as τ1→0 and (or) τ2→0. Precisely, we reveal that actually the indirect signal properties are strong enough to allow for a uniformly global-in-time convergence in terms of τ1=τ2=ϵ↘0 and τ1=1,τ2=ϵ↘0 for ∫Ωnϵ(⋅,0)<4π, which correspond to the classical Keller-Segel systems with direct signal production, and of τ1=ϵ↘0,τ2=1 for suitable small initial data. As a byproduct, we show that for any p≥2, the Lp-norm of nϵ blows up if ∫Ωnϵ(⋅,0)∈(4π,∞)∖4πN, which improves the existing results that the solution is unbounded in the sense of L∞-norm of nϵ.

Pervasiveness of the breakdown of self-interacting vector field theories

Various groups recently argued that self-interacting vector field theories lack a well-defined time evolution when the field grows to large amplitudes, which has drastic consequences for models in gravity and high energy theory. Such field amplitudes can be a result of an external driving mechanism, or occur intrinsically, due to large values of the field and its derivatives in the initial data. This brings a natural question: Is small amplitude initial data guaranteed to evolve indefinitely in these theories in the absence of an outside driving term? We answer this question in the negative, demonstrating that arbitrarily low amplitude initial data can still lead to the breakdown of the theory. Namely, ingoing spherically symmetric wave packets in more than one spatial dimensions grow as an inverse power of the radius, and their amplitudes can generically reach high enough values where time evolution ceases to exist. This simple example further establishes the pervasiveness of the pathology of self-interacting vector field theories.

Global existence for the 2D anisotropic Bénard equations with partial variable viscosity

The current paper aims to study the two‐dimensional (2D) Bénard equations with variable‐viscosity, where we obtain the global existence of a unique weak solution to this system without any assumptions of small initial data.

Global Sobolev regular solution for Boussinesq system

Abstract This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω ≔ R 2 × ( 0 , R ) \Omega := {{\mathbb{R}}}^{2}\times \left(0,R) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( Ω ) \left({\bf{v}},\rho )\in {H}^{s}\left(\Omega )\times {{\mathbb{H}}}^{s}\left(\Omega ) with the small initial data in the Sobolev space H s + 2 ( Ω ) × H s + 2 ( Ω ) {H}^{s+2}\left(\Omega )\times {{\mathbb{H}}}^{s+2}\left(\Omega ) . Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent s s can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.

Vanishing limit for the three-dimensional incompressible Phan-Thien–Tanner system

This paper focuses on the vanishing limit problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) system, which is commonly used to describe the dynamic properties of polymeric fluids. Our purpose is to show the relation of the PTT system to the well-known Oldroyd-B system (with or without damping mechanism). The suitable a priori estimates and global existence of strong solutions are established for the PTT system with small initial data. Taking advantage of uniform energy and decay estimates for the PTT system with respect to time $t$ and coefficients $a$ and $b$, then allows us to justify in particular the vanishing limit for all time. More precisely, we prove that the solution $(u,\,\tau )$ of PTT system with $0\leq b\leq Ca$ converges globally in time to some limit $(\widetilde {u},\,\widetilde {\tau })$ in a suitable Sobolev space when $a$ and $b$ go to zero simultaneously (or, only $b$ goes to zero). We may check that $(\widetilde {u},\,\widetilde {\tau })$ is indeed a global solution of the corresponding Oldroyd-B system. In addition, a rate of convergence involving explicit norm will be obtained. As a byproduct, similar results are also true for the local a priori estimates in large norm.

A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part I: Structures and well-posedness

We consider a model, which we named the Kakinuma model, for interfacial gravity waves. As is well known, the full model for interfacial gravity waves has a variational structure whose Lagrangian is an extension of Luke’s Lagrangian for surface gravity waves, that is, water waves. The Kakinuma model is a system of Euler–Lagrange equations for approximate Lagrangians, which are obtained by approximating the velocity potentials in the Lagrangian for the full model. In this paper we first analyze the linear dispersion relation for the Kakinuma model and show that the dispersion curves highly fit that of the full model in the shallow water regime. We then analyze the linearized equations around constant states and derive a stability condition, which is satisfied for small initial data when the denser water is below the lighter water. We show that the initial value problem is in fact well posed locally in time in Sobolev spaces under the stability condition, the noncavitation assumption, and intrinsic compatibility conditions, in spite of the fact that the initial value problem for the full model does not have any stability domain so that its initial value problem is ill posed in Sobolev spaces. Moreover, it is shown that the Kakinuma model enjoys a Hamiltonian structure and has conservative quantities: mass, total energy, and in the case of a flat bottom, momentum.

Stochastic Lagrangian perturbation of Lie transport and applications to fluids

In this paper, we propose a novel stochastic Lagrangian formulation of dissipatively perturbed Lie transport, which is based on the statistical generalized Cauchy invariants equation. This formulation consists of, first, finding a convenient Lagrangian formulation of the Lie transport equation involving particle trajectories, for instance the backward generalized Cauchy invariants equation (a Lagrangian formulation for Lie-advected exact p-forms, which is the Hodge dual of a generalization of the Cauchy vorticity formula), and second, performing a stochastic perturbation of the velocity-driven particle trajectories by adding to them white noises. Finally, using Itô’s calculus, an ensemble average of the stochastically perturbated generalized Cauchy invariants equation allows us to obtain the Lie transport equation perturbed by a deterministic potentially dissipative term given by the sum of squares of some Lie derivative operators. A remarkable property of this equation is that it satisfies a statistical Kelvin–Helmholtz theorem of conservation of circulation and flux. These results are obtained on periodic Euclidean spaces as well as on smooth closed Riemannian manifolds. In particular, we recover and thus generalize the Constantin–Iyer results on the stochastic Lagrangian formulation of the incompressible Navier–Stokes equations to a larger class of (deterministic) dissipative PDEs. A first application of this stochastic Lagrangian formulation is the derivation of new Lagrangian formulations for non-ideal (dissipative) hydrodynamic and magnetohydrodynamic models on flat and curved spaces, and in particular we obtain stochastic-Lagrangian incompressible extended MHD equations. As a second application, we use this new stochastic Lagrangian formulation to study the local well-posedness, the non-resistive limit and the global existence of classical solutions for the non-ideal incompressible extended MHD on the flat torus. Global-in-time existence is proved for small magnetic Reynolds numbers, that is, either for small initial data or large resistivity.

Well-posedness of the Cauchy problem for the kinetic DNLS on T

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus [Formula: see text] for [Formula: see text], where the constants [Formula: see text] are such that [Formula: see text] and [Formula: see text], and [Formula: see text] denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces [Formula: see text] for [Formula: see text]. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when [Formula: see text], cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in [Formula: see text], [Formula: see text]. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term [Formula: see text], in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.

Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime

We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent p satisfies 2≤p<1+2, we establish the sharp upper bound of lifespan estimate, while for the most delicate critical power p=1+2, we show that the lifespan satisfiesT(ε)≤exp⁡(Cε−(2+2)), the optimality of which remains to be proved. The key novelty is that the compact support of the initial data can be close to the event horizon. By combining the global existence result for p>1+2 obtained by Lindblad et al. (2014) [25], we then give a positive answer to the interesting question posed by Dafermos and Rodnianski (2005) [22, the end of the first paragraph in page 1151]: p=1+2 is exactly the critical power of p separating stability and blow-up.