arXiv subjects

Kaijian Sha

Publications and source records attributed to Kaijian Sha.

8 recordsLinked to original sources

arXiv2026

Asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones

We study the asymptotic long-time behavior of Darcy--Boussinesq convection in layered porous media with narrow transition zones in the material properties. As the transition-layer width tends to zero, we prove the upper semi-continuous convergence of the global attractor, invariant measure, and Nusselt number to their counterparts in the limiting sharp-interface model. We also show that the global attractors have finite fractal dimensions, with an explicit upper bound uniform in the transition-layer width. The analysis combines a carefully designed background temperature/contaminant profile together with a novel choice of phase space that ensures global well-posedness of the model and asymptotic compactness of the solution semigroup, and a new interpolation inequality. The phase space is associated with fractional powers of the principal elliptic operator with discontinuous coefficients. These results provide a rigorous long-time validation of the sharp-interface Darcy--Boussinesq model and extend our earlier finite-time convergence theory (H. Dong and X. Wang, SIAM J. Appl. Math. 85 (2025), 1621--1642) to the long-time regime.

math.AP↗

arXiv2026

Vanishing permeability limit of convection in multilayer porous media

We analyze the asymptotic behavior of the Boussinesq-Darcy system describing convection in layered porous media in the limit where the permeability of one layer tends to zero. We show that the limiting dynamics are governed by the Boussinesq-Darcy model with an impermeable layer, both in terms of convergence of solutions in L2 on finite time intervals and convergence of the corresponding global attractors. This limit is singular, as the pressure equation becomes degenerate when the permeability vanishes in part of the domain, resulting in a loss of uniqueness of the pressure in the impermeable layer. This difficulty is resolved by combining uniform estimates in the permeable layers with refined control of the pressure equation in the vanishing-permeability layer. The results provide a rigorous description of the zero-permeability limit in layered porous-media convection models.

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arXiv2025

Vanishing layer thickness limit of convection in multilayer porous media

Within the Darcy-Boussinesq framework for convection in multilayered porous media, we investigate the singular limit in which the thickness of one layer tends to zero. We establish that the solution of the full system converges to that of the corresponding limiting model with one fewer layer. The convergence is established in two complementary senses: (i) strong $L^{2}$-convergence over arbitrary finite time intervals, and (ii) upper semi-continuity of the global attractors describing the large-time asymptotic behavior.

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arXiv2024

On the asymptotic behavior of solutions to the steady Navier-Stokes system in two-dimensional channels

In this paper, we investigate the incompressible steady Navier-Stokes system with no-slip boundary condition in a two-dimensional channel. Given any flux, the existence of solutions is proved as long as the width of cross-section of the channel grows more slowly than the linear growth. Furthermore, if the flux is suitably small, the solution is unique even when the width of the channel is unbounded. Finally, based on the estimate of Dirichlet norm on the truncated domain, one could obtain the pointwise decay rate of the solution for arbitrary flux.

math.AP↗

arXiv2023

Uniqueness and uniform structural stability of Poiseuille flows with large fluxes in two-dimensional strips

In this paper, we prove the uniform nonlinear structural stability of Poiseuille flows with suitably large flux for the steady Navier-Stokes system in a two-dimensional strip with arbitrary period. Furthermore, the well-posedness theory for the Navier-Stokes system is also proved even when the $L^2$-norm of the external force is large. In particular, if the vertical velocity is suitably small where the smallness is independent of the flux, then Poiseuille flow is the unique solution of the steady Navier-Stokes system in the periodic strip. The key point is to establish uniform a priori estimates for the corresponding linearized problem via the boundary layer analysis, where we explore the particular features of odd and even stream functions. The analysis for the even stream function is new, which not only generalizes the previous study for the symmetric flows in \cite{Rabier1}, but also provides an explicit relation between the flux and period.

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arXiv2022

On the Leray problem for steady flows in two-dimensional infinitely long channels with slip boundary conditions

In this paper, we investigate the Leray problem for steady Navier-Stokes system under full slip boundary conditions in a two dimensional channel with straight outlets. The existence of solutions with arbitrary flux in a general channel with slip boundary conditions is established, which tend to the shear flows at far fields. Furthermore, if the flux is suitably small, the solutions are proved to be unique. One of the crucial ingredients is to construct an appropriate flux carrier and to show a Hardy type inequality for flows with full slip boundary conditions.

math.AP↗

arXiv2022

On the Steady Navier-Stokes system with Navier slip boundary conditions in two-dimensional channels

In this paper, we investigate the incompressible steady Navier-Stokes system with Navier slip boundary condition in a two-dimensional channel. As long as the width of cross-section of the channel grows more slowly than the linear growth, the existence of solutions with arbitrary flux is established. Furthermore, if the flux is suitably small, the solution is unique even when the width of the channel is unbounded, and approaches to the shear flows at far field where the channels tend to be straight at far fields. One of the major difficulties for the analysis on flows with Navier boundary conditions is that the tangential velocity may not be zero on the boundary so that we have to study the behavior of solutions near the boundary carefully. The crucial ingredients of analysis include the construction of an appropriate flux carrier, and the detailed analysis for the flow behavior near boundary via combining a Hardy type inequality for normal component of velocity and the divergence free property of the velocity.

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arXiv2020

Uniform structural stability and uniqueness of Poiseuille flows in a two dimensional periodic strip

In this paper, we prove the uniform nonlinear structural stability of Poiseuille flows with arbitrarily large flux for the Navier-Stokes system in a two dimensional periodic strip when the period is not large. The key point is to establish the a priori estimate for the associated linearized problem via the careful analysis for the associated boundary layers. Furthermore, the well-posedness theory for the Navier-Stokes system is also proved even when the external force is large in $L^2$. Finally, if the vertical velocity is suitably small where the smallness is independent of the flux, then Poiseuille flow is the unique solution of the steady Navier-Stokes system in the periodic strip.

math.AP↗