arXiv subjects

Shijin Ding

Publications and source records attributed to Shijin Ding.

At least 19 recordsLinked to original sources

arXiv2026

Global strong solutions for 1D compressible Navier-Stokes/Cahn-Hilliard equations with vacuum

In this paper, we study the initial-boundary value problem of the 1D compressible Navier--Stokes/Cahn--Hilliard system with vacuum. We establish the global existence and uniqueness of strong solutions to this initial-boundary value problem. No any initial compatibility conditions are required via time weighted techniques, which leads to a loss of regularity near the initial time. Therefore, the uniqueness of solutions obtained in this paper is even more challenging. To address this issue, we establish refined growth estimates and singular-in-time weighted energy estimates that induce a Gronwall-type structure, which ultimately allows us to close the uniqueness proof in Eulerian coordinates without passing to Lagrangian coordinates.

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arXiv2025

Global well-posedness of 2D incompressible MHD equations without magnetic diffusion

In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in \cite{Ren,TZhang}, under the assumption that initial data is close to equilibrium states with nontrivial magnetic field, and the perturbation is small in some suitable spaces, say for instance, the Sobolev spaces with negative exponents. It leads to an interesting open question: Can one establish the global existence of classical solutions without the extra help from Sobolev spaces with negative exponents like its counterparts of ideal MHD ( i.e. without viscosity and magnetic diffusion), and fully dissipative MHD (i.e. with both viscosity and magnetic diffusion)? This paper offers an affirmative answer to this question. In fact, we will establish the existence of a global unique solution for initial perturbations being small in $H^2(\mathbb{R}^2)$. The key idea is further exploring the structure of system, using dispersive effects of Alfv\'{e}n waves in the direction which is transversal to the dissipation favorable direction. This motivates our key strategy to treat the wildest nonlinear terms as an artificial linear term. These observations help us to construct some interesting quantities which improves the nonlinearity order for the wildest terms, and to control them by terms with better properties.

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arXiv2024

Stability for the 2-D plane Poiseuille flow in finite channel

In this paper, we study the stability for 2-D plane Poiseuille flow $(1-y^2,0)$ in a channel $\mathbb{T}\times (-1,1)$ with Navier-slip boundary condition. We prove that if the initial perturbation for velocity field $u_0$ satisfies that $\|u_0\|_{H^{\frac{7}{2}+}} \leq ε_1 ν^{2/3}$ for some suitable small $0<ε_1 \ll 1$ independent of viscosity coefficient $ν$, then the solution to the Navier-Stokes equations is global in time and does not transit from the plane Poiseuille flow. This result improves the result of \cite{DL1} from $3/4$ to $2/3$.

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arXiv2024

Nonlinear stability for 3-D plane Poiseuille flow in a finite channel

In this paper, we study the nonlinear stability for the 3-D plane Poiseuille flow $(1-y^2,0,0)$ at high Reynolds number $Re$ in a finite channel $\mathbb{T}\times [-1,1 ]\times \mathbb{T}$ with non-slip boundary condition. We prove that if the initial velocity $v_0$ satisfies $\|v_0-(1-y^2,0,0)\|_{H^{4}}\leq c_0 Re^{-\frac{7}{4}}$ for some $c_0>0$ independent of $Re$, then the solution of 3-D Naiver-Stokes equations is global in time and does not transit away from the plane Poiseuille flow. To our knowledge, this is the first nonlinear stability result for the 3-D plane Poiseuille flow and the transition threshold is accordant with the numerical result by Lundbladh et al. \cite{LHR}.

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arXiv2023

Well-posedness of Navier-Stokes/Cahn-Hilliard equations modeling the dynamics of contact line in a channel

In this paper, we study the contact line problem in a channel. Precisely, we consider the incompressible Navier-Stokes/Cahn-Hilliard system with eneralized Navier boundary condition and relaxation boundary condition in a channel, which is the phase field model for the moving contact line problem in fluid mechanics. We establish the existence and uniqueness of local-in-time strong solution to this initial boundary value problem in 2D. To our knowledge, this is the first result to give the local-in-time well-posedness of Navier-Stokes/Cahn-Hilliard system with generalized Navier boundary condition and relaxation boundary condition. This result provides a rigorous mathematical analysis to confirm that the physical and numerical results by Qian-Wang-Sheng [Phys. Rev. E 68 (2003), 016306, 1-15; J. Fluid Mech. 564 (2006), 333-360] are well-posed and reasonable.

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arXiv2022

Global-in-$x$ stability of Prandtl layer expansions for steady magnetohydrodynamics flows over a moving plate

In this paper, we obtain the global-in-$x$ Sobolev stability of Prandtl layer expansions for 2-D steady incompressible MHD flows with shear outer ideal MHD flows $(1,0,σ,0)$ ($σ\geq 0$) on a moving plate. It is worth noticing that in the Sobolev sense, the degeneracy of the tangential magnetic field in our result is allowed, which is much different from both the unsteady case in [25,26,27] and the steady case in [3,4].

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arXiv2022

Stability analysis for the incompressible Navier-Stokes equations with Navier boundary conditions

This paper concerns the instability and stability of the trivial steady states of the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then nonlinear asymptotic stability holds true, otherwise, there is a sharp critical viscosity, which distinguishes the nonlinear stability from instability.

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arXiv2021

Strong Solutions for 1D Compressible Navier-Stokes/Allen-Cahn System with Phase Variable Dependent Viscosity

This paper is concerned with a non-isentropic compressible Navier-Stokes/Allen-Cahn system with phase variable dependent viscosity $η(χ)=χ^α$ and temperature dependent heat-conductivity $κ(θ)=θ^β$. We show the global existence of strong solutions under some assumptions on growth exponent $α$ and initial data. It is worth noting that the initial data could be large if $α\ge0$ is small, and the growth exponent $β>0$ can be arbitrary large.

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arXiv2020

Stability of the boundary layer expansion for the 3D plane parallel MHD flow

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic (MHD) flow with no-slip boundary condition of velocity and perfectly conducting wall for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm $L^\infty(H^1)$. In addition, the similar convergence results are also obtained under the case with uniform magnetic fields. This implies the stabilizing effects of magnetic fields. Besides, the higher-order expansion is also considered.

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arXiv2020

Validity of Prandtl layer theory for steady magnetohydrodynamics over a moving plate with nonshear outer ideal MHD flows

In this paper, we validate the boundary layer theory for 2D steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ under the assumption of a moving boundary at $\{Y=0\}$. The validity of the boundary layer expansion and the convergence rates are established in Sobolev sense. We extend the results for the case with the shear outer ideal MHD flows [3] to the case of the nonshear flows.

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arXiv2020

Enhanced dissipation and transition threshold for the 2-D plane Poiseuille flow via resolvent estimate

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow $(1-y^2,0)$ in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate $e^{-c\sqrtνt}$. As an application, we prove that if the initial perturbation of vortiticy satisfies $$\|ω_0\|_{L^2}\leq c_0ν^{\frac{3}{4}},$$ for some small constant $c_0>0$ independent of the viscosity $ν$, then the solution dose not transition away from the Poiseuille flow for any time.

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arXiv2020

Rayleigh-Taylor instability for nonhomogeneous incompressible fluids with Navier-slip boundary conditions

This paper is concerned with the Rayleigh-Taylor instability for the nonhomogeneous incompressible Navier-Stokes equations with Navier-slip boundary conditions around a steady-state in an infinite slab, where the Navier-slip coefficients do not have defined sign and the slab is horizontally periodic. Motivated by [18], we extend the result from Dirichlet boundary condition to Navier-slip boundary conditions. Our results indicate the factor that "heavier density with increasing height" still plays a key role in the instability under Navier-slip boundary conditions.

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arXiv2020

Stability for two-dimensional plane Couette flow to the incompressible Navier-Stokes equations with Navier boundary conditions

This paper concerns with the stability of the plane Couette flow resulted from the motions of boundaries that the top boundary $Σ_1$ and the bottom one $Σ_0$ move with constant velocities $(a,0)$ and $(b,0)$, respectively. If one imposes Dirichlet boundary condition on the top boundary and Navier boundary condition on the bottom boundary with Navier coefficient $α$, there always exists a plane Couette flow which is exponentially stable for nonnegative $α$ and any positive viscosity $μ$ and any $a, b \in \mathbb{R}$, or, for $α<0$ but viscosity $μ$ and the moving velocities of boundaries $(a,0), (b,0)$ satisfy some conditions stated in Theorem 1.1. However, if we impose Navier boundary conditions on both boundaries with Navier coefficients $α_0$ and $α_1$, then it is proved that there also exists a plane Couette flow (including constant flow or trivial steady states) which is exponentially stable provided that any one of two conditions on $α_0,α_1$, $a, b$ and $μ$ in Theorem 1.2 holds. Therefore, the known results for the stability of incompressible Couette flow to no-slip (Dirichlet) boundary value problems are extended to the Navier boundary value problems.

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arXiv2020

Verification of Prandtl boundary layer ansatz for the steady electrically conducting fluids with a moving physical boundary

In this paper, we are concerned with the validity of Prandtl boundary layer expansion for the solutions to two dimensional (2D) steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ with a moving flat boundary $\{Y=0\}$. As a direct consequence, even though there exist strong boundary layers, the inviscid type limit is still established for the solutions of 2D steady viscous incompressible MHD equations in Sobolev spaces provided that the following three assumptions hold: the hydrodynamics and magnetic Reynolds numbers take the same order in term of the reciprocal of a small parameter $ε$, the tangential component of the magnetic field does not degenerate near the boundary and the ratio of the strength of tangential component of magnetic field and tangential component of velocity is suitably small. And the error terms are estimated in $L^\infty$ sense.

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arXiv2020

Inviscid Limit for the Free-Boundary problems of MHD Equations with or without Surface Tension

In this paper, we investigate the convergence rates of inviscid limits for the free-boundary problems of the incompressible magnetohydrodynamics (MHD) with or without surface tension in $\mathbb{R}^3$, where the magnetic field is identically constant on the surface and outside of the domain. First, we establish the vorticity, the normal derivatives and the regularity structure of the solutions, and develop a priori co-norm estimates including time derivatives by the vorticity system. Second, we obtain two independent sufficient conditions for the existence of strong vorticity layers: (I) the limit of the difference between the initial MHD vorticity of velocity or magnetic field and that of the ideal MHD equations is nonzero. (II) The cross product of tangential projection on the free surface of the ideal MHD strain tensor of velocity or magnetic field with the normal vector of the free surface is nonzero. Otherwise, the vorticity layer is weak. Third, we prove high order convergence rates of tangential derivatives and the first order normal derivative in standard Sobolev space, where the convergence rates depend on the ideal MHD boundary value.

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arXiv2019

Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flow of nonhomogeneous incompressible Navier-Stokes equations. The convergence for the density and velocity are shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the $L^\infty(H^1)$ norm. It is mentioned that the mathematical validity of the Prandtl boundary layer theory for nonlinear plane parallel flow is generalized to the nonhomogeneous case.

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arXiv2019

Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain

This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.

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arXiv2018

Symmetrical Prandtl boundary layer expansions of steady Navier-Stokes equations on bounded domain

This paper is concerned with the validity of the Prandtl boundary layer theory in the inviscid limit of the steady incompressible Navier-Stokes equations, which is an extension of the pioneer paper (Y. Guo et al., 2017, Ann. PDE) from a domain of $[0,L]\times\mathbb{R}_+$ to $[0,L]\times[0,2]$. Under the symmetry assumption, we establish the validity of the Prandtl boundary layer expansions and the error estimates. The convergence rate as $\e\rightarrow 0$ is also given.

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