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Tong Yang - One of the best experts on this subject based on the ideXlab platform.
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magnetic effects on the solvability of 2d mhd boundary layer equations without resistivity in sobolev spaces
Journal of Functional Analysis, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:Abstract In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary Condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the Monotonicity Condition on the velocity field. Moreover, we show that if the tangential magnetic field of shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Gevrey function space provided that the initial velocity shear flow is non-degenerately critical at the same point.
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magnetic effects on the solvability of 2d mhd boundary layer equations without resistivity in sobolev spaces
arXiv: Analysis of PDEs, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary Condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the Monotonicity Condition on the velocity field. Moreover, we show that if the tangential magnetic field shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point.
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a note on the ill posedness of shear flow for the mhd boundary layer equations
Science China-mathematics, 2018Co-Authors: Chengjie Liu, Feng Xie, Tong YangAbstract:For the two-dimensional Magnetohydrodynamics (MHD) boundary layer system, it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolevspaces and high Reynolds number limits without any Monotonicity Condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in thetangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by Gerard-Varet and Dormy (2010). This partially shows the necessity of thenon-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.
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Well-posedness of The Prandtl Equation in Sobolev Spaces
2012Co-Authors: Radjesvarane Alexandre, Ya-guang Wang, Tong YangAbstract:We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a Monotonicity Condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.
Chengjie Liu - One of the best experts on this subject based on the ideXlab platform.
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magnetic effects on the solvability of 2d mhd boundary layer equations without resistivity in sobolev spaces
Journal of Functional Analysis, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:Abstract In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary Condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the Monotonicity Condition on the velocity field. Moreover, we show that if the tangential magnetic field of shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Gevrey function space provided that the initial velocity shear flow is non-degenerately critical at the same point.
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magnetic effects on the solvability of 2d mhd boundary layer equations without resistivity in sobolev spaces
arXiv: Analysis of PDEs, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary Condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the Monotonicity Condition on the velocity field. Moreover, we show that if the tangential magnetic field shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point.
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a note on the ill posedness of shear flow for the mhd boundary layer equations
Science China-mathematics, 2018Co-Authors: Chengjie Liu, Feng Xie, Tong YangAbstract:For the two-dimensional Magnetohydrodynamics (MHD) boundary layer system, it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolevspaces and high Reynolds number limits without any Monotonicity Condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in thetangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by Gerard-Varet and Dormy (2010). This partially shows the necessity of thenon-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.
Feng Xie - One of the best experts on this subject based on the ideXlab platform.
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magnetic effects on the solvability of 2d mhd boundary layer equations without resistivity in sobolev spaces
Journal of Functional Analysis, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:Abstract In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary Condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the Monotonicity Condition on the velocity field. Moreover, we show that if the tangential magnetic field of shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Gevrey function space provided that the initial velocity shear flow is non-degenerately critical at the same point.
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magnetic effects on the solvability of 2d mhd boundary layer equations without resistivity in sobolev spaces
arXiv: Analysis of PDEs, 2020Co-Authors: Chengjie Liu, Tong Yang, Feng Xie, Dehua WangAbstract:In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary Condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the Monotonicity Condition on the velocity field. Moreover, we show that if the tangential magnetic field shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point.
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a note on the ill posedness of shear flow for the mhd boundary layer equations
Science China-mathematics, 2018Co-Authors: Chengjie Liu, Feng Xie, Tong YangAbstract:For the two-dimensional Magnetohydrodynamics (MHD) boundary layer system, it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolevspaces and high Reynolds number limits without any Monotonicity Condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in thetangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by Gerard-Varet and Dormy (2010). This partially shows the necessity of thenon-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.
Chengming Huang - One of the best experts on this subject based on the ideXlab platform.
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projected euler maruyama method for stochastic delay differential equations under a global Monotonicity Condition
Applied Mathematics and Computation, 2020Co-Authors: Chengming HuangAbstract:Abstract In this paper, we investigate a projected Euler-Maruyama method for stochastic delay differential equations with variable delay under a global Monotonicity Condition. This Condition admits some equations with highly nonlinear drift and diffusion coefficients. We appropriately generalize the idea of C-stability and B-consistency given by Beyn et al. (2016) to the case with delay. Moreover, the method is proved to be convergent with order one-half in a succinct way. Finally, some numerical examples are included to support our theoretical results.
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compensated projected euler method for stochastic differential equations with jumps under global Monotonicity Condition
arXiv: Numerical Analysis, 2018Co-Authors: Chengming HuangAbstract:This paper presents and analyzes the compensated projected Euler-Maruyama method for stochastic differential equations with jumps under a global Monotonicity Condition. Compared with existing Conditions, this Condition allows the jump-diffusion coefficient to be growth superlinearly. Moreover, the method is proved to be convergent with strongly order $\frac{1}{2}$ on the discrete time level. Finally, some numerical experiments are carried out to confirm the theoretical results.
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projected euler method for stochastic delay differential equation under a global Monotonicity Condition
arXiv: Numerical Analysis, 2018Co-Authors: Chengming HuangAbstract:This paper investigates projected Euler-Maruyama method for stochastic delay differential equations under a global Monotonicity Condition. This Condition admits some equations with highly nonlinear drift and diffusion coefficients. We appropriately generalized the idea of C-stability and B-consistency given by Beyn et al. [J. Sci. Comput. 67 (2016), no. 3, 955-987] to the case with delay. Moreover, the method is proved to be convergent with order $\frac{1}{2}$ in a succinct way. Finally, some numerical examples are included to illustrate the obtained theoretical results.
Anis Matoussi - One of the best experts on this subject based on the ideXlab platform.
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sobolev solution for semilinear pde with obstacle under Monotonicity Condition
Electronic Journal of Probability, 2008Co-Authors: Anis MatoussiAbstract:We prove the existence and uniqueness of Sobolev solution of a semilinear PDE's and PDE's with obstacle under Monotonicity Condition. Moreover we give the probabilistic interpretation of the solutions in term of Backward SDE and reflected Backward SDE respectively
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sobolev solution for semilinear pde with obstacle under Monotonicity Condition
arXiv: Probability, 2007Co-Authors: Anis MatoussiAbstract:We prove the existence and uniqueness of the solution of a semilinear PDE's and also PDE's with obstacle under Monotonicity Condition. Moreover we give the probabilistic interpretation of the Sobolev's solutions in term of Backward SDE and reflected Backward SDE respectively.