Global Well-posedness Of Strong Solutions Research Articles

Global well-posedness of strong solutions to the two-dimensional inhomogeneous biaxial nematic liquid crystal flow with vacuum

This paper considers the inhomogeneous biaxial nematic liquid crystal flow in a smooth bounded domain Ω⊂R2, where the velocity u and the orthogonal unit vector fields (m,n) admit the Dirichlet and Neumann boundary condition, respectively. By applying piecewise estimate and continuity method, we get the global existence of strong solutions, provided that the basic energy is suitably small. Our result may be regarded as an extension and improvement of Gong-Lin (2022) and Li-Liu-Zhong (2017) to the Neumann boundary condition, where the initial vacuum is allowed. Some new techniques are developed in order to deal with integral estimates caused by the boundary condition, and more complicated model.

On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

<p>In this paper, we considered the global well-posedness of strong solutions to the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible micropolar fluids with density-dependent viscosity and vacuum. Based on the energy method, some key a priori exponential decay-in-time rates of strong solutions are obtained. As a result, the existence and large-time asymptotic behavior of strong solutions in the whole space $ \mathbb{R}^3 $ are established, provided that the initial mass is sufficiently small. Note that this result is proven without any compatibility conditions.</p>

Global well-posedness of strong solutions to the primitive equations without vertical diffusivity

This paper is concerned with the initial–boundary value problem of the three-dimensional primitive equations for oceanic dynamics. The global existence of strong solutions to the primitive equations without vertical diffusivity is proved under the assumption of initial data (v0,T0)∈H1.

Global classical solutions for three-dimensional compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in bounded domains

In this paper, we study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a finite number of two-dimensional connected components. We derive the global existence and uniqueness of classical solutions provided that the initial total energy is suitably small. Our result generalizes the Cauchy problems of compressible Navier–Stokes equations with Coulomb force [S. Liu, X. Xu and J. Zhang, Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier–Stokes–Poisson equations subject to large oscillations and non-flat doping profile, J. Differ. Equ. 269 (2020) 8468–8508] and compressible MHD equations [H. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal. 45 (2013) 1356–1387] to the case of bounded domains although tackling many surface integrals caused by the slip boundary condition are complex. The main ingredient of this paper is to overcome the strong nonlinearity caused by Coulomb force, magnetic field, and rotation effect of micro-particles by applying piecewise-estimate method and delicate analysis based on the effective viscous fluxes involving velocity and micro-rotational velocity.

Global strong solutions of three-dimensional heat conducting incompressible magnetohydrodynamic equations with vacuum

This paper is devoted to studying global existence and large time behavior of strong solutions of the three-dimensional heat conducting incompressible magnetohydrodynamic flows with initial vacuum and a vacuum far field. Under the scaling invariant quantity (1+‖ρ0‖∞)2(‖ρ0u0‖22+‖b0‖22)(‖∇u0‖22+‖∇b0‖22) is suitably small, the global well-posedness of strong solution to the Cauchy problem is proved. Here, the explicit dependence on the initial norms is given and the smallness depends only on the known parameters in the system. This implies that the global well-posedness result is also valid for some large initial data. Furthermore, the algebraic decay rates of the global solution are obtained.

The Global Well-posedness of Strong Solutions to 2D MHD Equations in Lei-Lin Space

The Global Well-posedness of Strong Solutions to 2D MHD Equations in Lei-Lin Space

On the Rigorous Mathematical Derivation for the Viscous Primitive Equations with Density Stratification

In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.

Global strong solutions of 3D compressible isothermal magnetohydrodynamics

This paper establishes the global existence of strong solutions for the three-dimensional compressible isothermal magnetohydrodynamic (MHD) flows. It is essentially shown that for the regular data with small energy but possible large oscillations, the global well-posedness of strong solutions is established.

Global well-posedness to the compressible nematic liquid crystal flows with large oscillations and vacuum in 3D exterior domains

This paper aims at extending our previous work (Y. Liu and X. Zhong, arXiv:2204.06227) to the exterior problem. We show the global well-posedness of strong solutions to a simplified Ericksen–Leslie model of compressible nematic liquid crystal flows in a three-dimensional exterior domain when the initial total energy is sufficiently small. In order to obtain necessary a priori estimates on the boundary, some new analytic techniques are used.

Global well-posedness to the Cauchy problem of 2D nonhomogeneous Bénard system with large initial data and vacuum

This paper establishes the global well-posedness of strong solutions to the nonhomogeneous Bénard system with positive density at infinity in the whole space R2. We obtain the global existence and uniqueness of strong solutions for general large initial data. Our method relies on the dedicate energy estimates and a logarithmic interpolation inequality.

Global strong solutions to the Cauchy problem of the planar non-resistive magnetohydrodynamic equations with large initial data

In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced “transverse effective viscous flux” vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.

Global existence of strong solutions for a general incompressible Oldroyd-B system without damping mechanism

In this paper, we consider the three dimensional Cauchy problem for a general incompressible Oldroyd-B system without damping mechanism, and establish the global well-posedness of strong solutions.

Global Well-Posedness and Exponential Decay of 2D Nonhomogeneous Navier–Stokes and Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum

We establish global well-posedness of strong solutions for the nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and initial density allowing vanish in two-dimensional bounded domains. Applying delicate energy estimates and Desjardins’ interpolation inequality, we derive the global existence of a unique strong solution provided that \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}\) is suitably small. Moreover, we also obtain exponential decay rates of the solution. In particular, there is no need to impose some compatibility condition on the initial data despite the presence of vacuum. As a direct application, it is shown that the similar result also holds for the nonhomogeneous Navier–Stokes equations with density-dependent viscosity.

On non-resistive limit of 1D MHD equations with no vacuum at infinity

Abstract In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

Boundary Conditions and Polymeric Drag Reduction for the Navier–Stokes Equations

Reducing wall drag in turbulent pipe and channel flows is an issue of great practical importance. In engineering applications, end-functionalized polymer chains are often employed as agents to reduce drag. These are polymers which are floating in the solvent and attach (either by adsorption or through irreversible chemical binding) at one of their chain ends to the substrate (wall). We propose a PDE model to study this setup in the simple setting where the solvent is a viscous incompressible Navier–Stokes fluid occupying the bulk of a smooth domain \(\Omega \subset {\mathbb {R}}^d\), and the wall-grafted polymer is in the so-called mushroom regime (inter-polymer spacing on the order of the typical polymer length). The microscopic description of the polymer enters into the macroscopic description of the fluid motion through a dynamical boundary condition on the wall-tangential stress of the fluid, something akin to (but distinct from) a history-dependent slip-length. We establish the global well-posedness of strong solutions in two-spatial dimensions and prove that the inviscid limit to the strong Euler solution holds with a rate. Moreover, the wall-friction factor \(\langle f\rangle \) and the global energy dissipation \(\langle \varepsilon \rangle \) vanish inversely proportional to the Reynolds number \(\mathbf{Re } \). This scaling corresponds to Poiseuille’s law for the friction factor \(\langle f\rangle \sim 1/\mathbf{Re } \) for laminar flow and thereby quantifies drag reduction in our setting. These results are in stark contrast to those available for physical boundaries without polymer additives modeled by, for example, no-slip conditions, where no such results are generally known even in two-dimensions.

Global well-posedness of strong solution to 2D MHD equations in critical Fourier-Herz spaces

In this paper, we study the Cauchy problem for the 2D MHD equations in critical Fourier-Herz space FB˙1,q−1(R2)∩L2(R2)(q∈[1,2]). Making use of Fourier localization method and Banach fixed point theorem, we established the global well-posedness of a strong solution with any initial data (u0,b0)∈FB˙1,q−1(R2)∩L2(R2).

Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum

We establish global well-posedness of strong solutions to the nonhomogeneous heat conducting magnetohydrodynamic equations with non-negative density on the whole space $${\mathbb {R}}^2$$ . More precisely, under compatibility conditions for the initial data, we show the global existence and uniqueness of strong solutions. Our method relies on delicate energy estimates and a logarithmic interpolation inequality. In particular, the initial data can be arbitrarily large.

Global well-posedness of strong solutions to the 2D nonhomogeneous incompressible primitive equations with vacuum

We prove the global well-posedness of the strong solution to the two-dimensional inhomogeneous incompressible primitive equations for initial data with small H12-norm, which also satisfies a natural compatibility condition. A logarithmic type Sobolev embedding inequality for the anisotropic Lx∞Lz2(Ω) norm is established to obtain the global in time a priori H1(Ω)∩W1,6(Ω) estimate of the density, which guarantee the local solution to be a global one.

Corrigendum to “Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile” [J. Differ. Equ. 269 (10) (2020) 8468–8508

Corrigendum to “Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile” [J. Differ. Equ. 269 (10) (2020) 8468–8508

On global well-posedness and decay of 3D Ericksen-Leslie system

<abstract><p>In this paper, the small initial data global well-posedness and time decay estimates of strong solutions to the Cauchy problem of 3D incompressible liquid crystal system with general Leslie stress tensor are studied. First, assuming that $ \|u_0\|_{\dot{H}^{\frac12+\varepsilon}}+\|d_0-d_*\|_{\dot{H}^{\frac32+\varepsilon}} $ ($ \varepsilon > 0) $ is sufficiently small, we obtain the global well-posedness of strong solutions. Moreover, the $ L^p $–$ L^2 $ ($ \frac32\leq p\leq2 $) type optimal decay rates of the higher-order spatial derivatives of solutions are also obtained. The $ \dot{H}^{-s} $ ($ 0\leq s < \frac12 $) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.</p></abstract>