Global Existence Of Strong Solutions Research Articles

Global strong solutions to the radially symmetric compressible MHD equations in 2D solid balls with arbitrary large initial data

In this paper, we prove the global existence of strong solutions to the two-dimensional compressible MHD equations with density dependent viscosity coefficients (known as Kazhikhov-Vaigant model) on 2D solid balls with arbitrary large initial smooth data where shear viscosity μ being constant and the bulk viscosity λ be a polynomial of density up to power β. The global existence of the radially symmetric strong solutions was established under Dirichlet boundary conditions for β > 1. Moreover, as long as β>max{1,γ+24}, the density is shown to be uniformly bounded with respect to time. This generalizes the previous result [Fan et al., Arch. Ration. Mech. Anal. 245, 239–278 (2022)] of the compressible Navier-Stokes equations on 2D bounded domains where they require β > 4/3 and also improves the result [Chen et al., SIAM J. Math. Anal. 54(3), 3817–3847 (2022)] of of compressible MHD equations on 2D solid balls.

Global existence of strong solutions to the isentropic compressible nematic liquid crystal equations in a cuboid domain

In this paper, we study the initial-boundary value problem of three-dimensional compressible nematic liquid crystal equations in a cuboid domain. We prove that the strong solution exists globally in the time provided that both the initial L1-norm of the density ‖ρ0‖L1 and the initial L3-norm of the gradient of orientation field ‖∇d0‖L3 for nematic liquid crystal flow are small enough. The main tools of proving the global well-posedness are some time-weighted a priori estimates designed for the compressible nematic liquid crystal equations.

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 2D compressible fluid model with the horizontal thermal gradient effect

This article mainly investigates the existence of local strong solution to the Cauchy problem of the 2D compressible fluid model with the horizontal thermal gradient effect. By using delicate weighted energy estimates, the existence of a local strong solution is obtained with the initial density and the initial temperature decay not too slowly at infinity.

The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain

The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain

The global existence of strong solutions for a non-isothermal ideal gas system

The global existence of strong solutions for a non-isothermal ideal gas system

On the global stability of large Fourier mode for 3-D Navier-Stokes equations

In this paper, we first prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations with solenoidal initial data, which writes in the cylindrical coordinates is of the form: A(r,z)cos⁡Nθ+B(r,z)sin⁡Nθ, provided that N is large enough. In particular, we prove that the corresponding solution has almost the same frequency N for any positive time. The main idea of the proof is first to write the solution in trigonometrical series in θ variable and estimate the coefficients separately in some scale-invariant spaces, then we handle a sort of weighted sum of these norms of the coefficients in order to close the a priori estimate of the solution. Furthermore, we shall extend the above well-posedness result for initial data which is a linear combination of axisymmetric data without swirl and infinitely many large mode trigonometric series in the angular variable.

On the existence of global strong solutions to 1D bilayer shallow water model

On the existence of global strong solutions to 1D bilayer shallow water model

Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

Global well-posedness of 2D incompressible Navier–Stokes–Darcy flow in a type of generalized time-dependent porosity media

<p>This study investigates the global well-posedness of a coupled Navier–Stokes–Darcy model incorporating the Beavers–Joseph–Saffman–Jones interface boundary condition in two-dimensional Euclidean space. We establish the existence of global strong solutions for the system in both linear and nonlinear cases where porosity depends on pressure. When dealing with the time-dependent porous media, the primary challenge in obtaining closed prior estimates arises from the presence of complex, sharp interfaces. To address this issue, we employ the classical Trace Theorem. Such space-time variable coupled systems are crucial for understanding underground fluid flow.</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 well-posedness and large-time behavior of solutions to the 3D inviscid magneto-micropolar equations with damping

We first establish the existence of global strong solutions to the 3D inviscid incompressible magneto-micropolar equations with a velocity damping in Sobolev spaces H K ( K ⩾ 3 ) , where the H 3 norm of initial data is small, but its higher order derivatives could be large. Combining the H ˙ − s norm ( 0 ≤ s < 3 2 ) or B ˙ 2 , ∞ − s norm ( 0 < s ≤ 3 2 ) of the initial magnetic field is finite with some tricky interpolation estimates, we show ‖ ∇ n B ( t ) ‖ L 2 ≤ C 0 ( 1 + t ) − n 2 − 3 2 p + 3 4 and two faster decay rates for the velocity and the micro-rotation field ‖ ∇ n ( U , w ) ( t ) ‖ L 2 ≤ C 0 ( 1 + t ) − n 2 − 3 2 p + 1 4 , which are shown to be the usual L p − L 2 ( 1 ≤ p ≤ 2 ) type of the optimal time decay rates for 3D inviscid incompressible magneto-micropolar equations with damping. We then conclude the damping term contributes to weaken the assumption of initial condition and enhance the decay rate of velocity compared to the classical incompressible viscous magneto-micropolar equations. Meanwhile, for the 3D inviscid compressible magneto-micropolar fluid, the damping term also has the same effect on the decay rate of the velocity.

Global Existence of Strong Solutions and Serrin-Type Blowup Criterion for 3D Combustion Model in Bounded Domains

Global Existence of Strong Solutions and Serrin-Type Blowup Criterion for 3D Combustion Model in Bounded Domains

Remarks on strong global solutions of the [formula omitted]-equation

A bounded first-order derivative with respect to x of a solution to the b-equation, which encompasses the Camassa–Holm and Degasperis–Procesi equations, serves as a crucial condition that triggers a process leading to the establishment of bounds for derivatives of the solution up to an order determined by the regularity of the initial data. Consequently, this condition ensures the global existence of strong solutions in Sobolev spaces.

Global well-posedness and convergence to equilibrium for the Abels-Garcke-Grün model with nonlocal free energy

We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existence of global strong solutions in general two-dimensional bounded domains and their uniqueness when the initial datum is strictly separated from the pure phases. The key points are a novel well-posedness result of strong solutions to the nonlocal convective Cahn-Hilliard equation with singular potential and constant mobility under minimal integral assumption on the incompressible velocity field, and a new two-dimensional interpolation estimate for the L4(Ω) control of the pressure in the stationary Stokes problem. Secondly, we show that any weak solution, whose existence was already known, is globally defined, enjoys the propagation of regularity and converges towards an equilibrium (i.e., a stationary solution) as t→∞. Furthermore, we demonstrate the uniqueness of strong solutions and their continuous dependence with respect to general (not necessarily separated) initial data in the case of matched densities and unmatched viscosities (i.e., the nonlocal model H with variable viscosity, singular potential and constant mobility). Finally, we provide a stability estimate between the strong solutions to the nonlocal AGG model and the nonlocal Model H in terms of the difference of densities.

Optimal decay rates for the viscous two-phase model without constraints on transition to single-phase flow

In this paper, we study a free-boundary problem of the liquid-gas two-phase drift-flux model with constant viscosity, where the initial liquid and gas masses are assumed to be connected with vacuum continuously, and the pressure function takes a singular form. We utilize the flow map to reformulate the problem and prove the global existence and decay rates of strong solutions with general initial data that involve transition to pure single-phase points or regions. In particular, both the optimal decay rates of the mass functions and the decay estimates of the velocity towards its asymptotic profile as time goes to infinity are studied. As far as we are concerned, this is the first result on the global existence of strong solutions to the two-phase drift-flux model with singular pressure function and with general initial data that allow for transition to single-phase flow. It is also worth noting that our method in recovering the non-weighted estimates of the solution by utilizing the Poincaré type inequality can apply to the single-phase problem studied in Zeng (2015) [37], so as to derive the decay estimates for the velocity and relax the constraints on the initial data.

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.

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.

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.

On the strong solution of 3D non-isothermal Navier–Stokes–Cahn–Hilliard equations

In this paper, we consider the global existence of strong solutions of a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. In the diffuse interface model, the evolution of the velocity u is ruled by the Navier–Stokes system, while the order parameter φ representing the difference of the fluid concentration of the two fluids is assumed to satisfy a convective Cahn–Hilliard equation. The effects of the temperature are prescribed by a suitable form of heat equation. By using a refined pure energy method, we prove the existence of the global strong solution by assuming that ‖u0‖H32+‖φ0‖H42+‖θ0‖H32+‖φ02−1‖L22+‖θ0‖L1 is sufficiently small, and higher order derivatives can be arbitrarily large.