Global Well-posedness Of Classical Solutions Research Articles

Global Well-Posedness of Classical Solutions to the Compressible Navier–Stokes–Poisson Equations with Slip Boundary Conditions in 3D Bounded Domains

Global Well-Posedness of Classical Solutions to the Compressible Navier–Stokes–Poisson Equations with Slip Boundary Conditions in 3D Bounded Domains

Global strong solutions to the compressible magnetohydrodynamic equations with slip boundary conditions in 3D bounded domains

We deal with the barotropic compressible magnetohydrodynamic equations in three-dimensional (3D) bounded domain with slip boundary condition and vacuum. By a series of a priori estimates, especially the boundary estimates, we prove the global well-posedness of classical solution and the exponential decay rate to the initial-boundary-value problem of this system for the regular initial data with small energy but possibly large oscillations. The initial density of such a classical solution is allowed to contain vacuum states. Moreover, it is also shown that the oscillation of the density will grow unboundedly with an exponential rate when the initial state contains vacuum.

Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum

This paper is concerned with the Cauchy problem of Navier–Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in [Formula: see text]. For less regular data and weaker compatibility condition than those proposed by Cho–Kim [Existence results for viscous polytropic fluids with vacuum, J. Differ. Equ. 228 (2006) 377–411], we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is more regular. We also establish the global well-posedness of classical solutions with large oscillations and vacuum in the case when the initial total energy is suitably small. The exponential decay estimates of the global solutions are obtained.

Global Well-Posedness of Classical Solutions to the Cauchy Problem of Two-Dimensional Barotropic Compressible Navier--Stokes System with Vacuum and Large Initial Data

The Cauchy problem for the barotropic compressible Navier--Stokes equations on the whole two-dimensional space with vacuum as far field density is considered. When the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds, the global existence and uniqueness of strong and classical solutions is established. It should be remarked that there are no restrictions on the size of the data.

Global well-posedness of classical solution to the interactions between short-long waves with large initial data

We present the global existence and uniqueness of the classical solution to a system describing the interactions of short waves and long waves. This coupled system is formed by the compressible micropolar equations in Eulerian coordinates and the Schrödinger equation in Lagrangian coordinates. This paper establishes a sufficient frame condition to guarantee the well-posedness of the coupled system with large initial data in the two-dimensional torus.

On existence of global classical solutions to the 3D compressible MHD equations with vacuum

In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on mathbb{R}^{3}times [0, T] with any Tin (0, infty ), provided that the initial magnetic field in the L^{3}-norm and the initial density are suitably small. Note that the first result is obtained under the condition of rho _{0}in L^{gamma }cap W^{2, q} with qin (3, 6) and gamma in (1, 6). It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., rho _{0} notin L^{1}). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved.

Global regularity for the 2D micropolar Rayleigh–Bénard problem with partial dissipation

This paper is devoted to the study of the global regularity problem to the 2D micropolar Rayleigh–Bénard problem with only partial velocity dissipation and micro-rotational dissipation in R2. By fully exploiting the special structure of the system and using the delicate energy methods, we establish the global well-posedness of classical solutions for this system.

Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics

<p style='text-indent:20px;'>We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a <inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.</p>

Large time behavior and diffusion limit for a system of balance laws from chemotaxis in multi-dimensions

We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero.

Global classical solutions to two-dimensional chemotaxis-shallow water system

<p style='text-indent:20px;'>We consider the Cauchy problem of two-dimensional chemotaxis-shallow water system in the present paper. For regular initial data with small energy but possibly large oscillations, we prove the global well-posedness of classical solution. Then, we show the large-time behavior of the solution using the time-independent lower-order estimates as well.

On the global existence of classical solutions for compressible nematic liquid crystal flows with vacuum

This paper deals with the Cauchy problem of compressible nematic liquid crystal flows in the whole space $$\mathbb {R}^3$$. We show that if, in addition, the conservation law of the total mass is satisfied (i.e., $$\rho _0\in L^1$$), then the global existence theorem with small density and $$L^3$$-norm of the gradient of $$d_0$$ holds for any $$\gamma >1$$. It is worth mentioning that the initial total energy can be arbitrarily large and the initial vacuum is allowed. Thus, the result obtained particularly extends the one due to Li et al. (J Math Fluid Mech 20:2105–2145, 2018), where the global well-posedness of classical solutions with small energy was proved.

Optimal decay to the non-isentropic compressible micropolar fluids

In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropolar fluid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equilibrium state with rate $ (1+t)^{-3 / 4} $ in $ L^{2} $ norm and the micro-rotational velocity tends to the equilibrium state with the faster rate $ (1+t)^{-5 / 4} $ in $ L^{2} $ norm. The proof is based on the detailed analysis of the Green function and time-weighted energy estimates.

Global classical solutions to 1D full compressible Navier–Stokes equations with the Robin boundary condition on temperature

In this paper, we consider the initial boundary value problem for the one-dimensional Navier–Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature. There are few results until now about global existence of regular solutions to the full Navier–Stokes equations with the Robin boundary condition on temperature. By the analysis of the effect of boundary dissipation, we derive the global existence of classical solution to the corresponding initial boundary value problem with large initial data and vacuum. This result could be viewed as the first one on the global well-posedness of classical solutions to the full Navier–Stokes equations in a bounded domain with the Robin boundary condition on temperature.

Global regularity for the 2D magneto-micropolar equations with partial and fractional dissipation

This paper studies two cases of global regularity problems on the 2D magneto-micropolar equations with partial magnetic diffusion and fractional dissipation. For the first case the velocity field is ideal, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has fractional partial diffusion (−∂22βb1,−∂11βb2) with β>1. In the second case, the velocity has a fractional Laplacian dissipation (−Δ)αu with any α>0, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has partial diffusion (−∂22b1,−∂11b2). In two cases the global well-posedness of classical solutions is proved in this paper.

Global classical solutions of compressible isentropic Navier–Stokes equations with small density

This paper concerns the Cauchy problem of compressible isentropic Navier–Stokes equations in the whole space R3. First, we show that if ρ0∈Lγ∩H3, then the problem has a unique global classical solution on R3×[0,T] with any T∈(0,∞), provided the upper bound of the initial density is suitably small and the adiabatic exponent γ∈(1,6). If, in addition, the conservation law of the total mass is satisfied (i.e., ρ0∈L1), then the global existence theorem with small density holds for any γ>1. It is worth mentioning that the initial total energy can be arbitrarily large and the initial vacuum is allowed. Thus, the results obtained particularly extend the one due to Huang–Li–Xin (Huang et al., 2012), where the global well-posedness of classical solutions with small energy was proved.

Global classical solutions to the 3D Navier–Stokes–Korteweg equations with small initial energy

In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stress tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space [Formula: see text].

Global well-posedness of classical solutions to a fluid–particle interaction model in [formula omitted

We consider the Cauchy problem of a fluid–particle interaction model in R3, namely, compressible Navier–Stokes equations coupled with Smoluchowski equation. When the initial data (ρ0,u0,η0) is of small energy around steady state (ρ⋆,0,η⋆), the global well-posedness and large-time behavior of classical solutions are investigated. Vacuum is allowed.

Global well-posedness for systems of elasticity

This is a survey paper. We mainly revisit those highlighted results on the global well-posedness of classical solutions near equilibrium for systems ofcompressible and incompressible elasticity. Due to the fact that the methods and ideas used for elasticity are closely related to those used for quasilinear wave equations, we will also revisit some corresponding highlighted results on quasilinear wave equations. We will try to pinpoint the key difficulties of those problems and the key methods to overcome them, and present direct and simple proofs as many as we can.We will also raise some open questions and make a few comments to attract the attention of young researchers.

Global small solutions to a tropical climate model without thermal diffusion

We obtain the global well-posedness of classical solutions to a tropical climate model with only the dissipation of the first baroclinic model of the velocity (−ηΔv) under small initial data. This model is a modified version of the original system derived by Frierson-Majda-Pauluis in Frierson et al. [Commun. Math. Sci. 2, 591-626 (2004)]. The main difficulty is the absence of thermal diffusion. To overcome it, we exploit the structure of the equations coming from the coupled terms, dissipation term, and damp term. Then, we find the hidden thermal diffusion. In addition, based on the Littlewood-Paley theory, we establish a generalized commutator estimate, which may be applied to other partial differential equations.

On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.