This article gives a general model using specific periodic special functions, which is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the perio...This article gives a general model using specific periodic special functions, which is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of cells of the 3-Torus. Satisfying a divergence-free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which have finite time singularities can occur starting with the first derivative and higher with respect to time. The existence of a subspace of the solution space where v<sub>3</sub> is continuous and {C, y<sub>1</sub>, y<sub>1</sub><sup>2</sup>}, is linearly independent in the additive argument of the solution in terms of the Lambert W function, (y<sub>1</sub><sup>2</sup>=y<sub>2</sub>, C∈R) together with the condition v<sub>2</sub>=-2y<sub>1</sub>v<sub>1</sub>. On this subspace, the Biot Savart Law holds exactly [see Section 2 (Equation (13))]. Also on this subspace, an expression X (part of PNS equations) vanishes which contains all the expressions in derivatives of v<sub>1</sub> and v<sub>2</sub> and the forcing terms in the plane which are related as with the cancellation of all such terms in governing PDE. The y<sub>3</sub> component forcing term is arbitrarily small in ε ball where Weierstrass P functions touch the center of the ball both for inviscid and viscous cases. As a result, a significant simplification occurs with a v<sub>3 </sub>only governing PDE resulting. With viscosity present as v changes from zero to the fully viscous case at v =1 the solution for v<sub>3</sub> reaches a peak in the third component y<sub>3</sub>. Consequently, there exists a dipole which is not centered at the center of the cell of the Lattice. Hence since the dipole by definition has an equal in magnitude positive and negative peak in y<sub>3</sub>, then the dipole Riemann cut-off surface is covered by a closed surface which is the sphere and where a given cell of dimensions [-1, 1]<sup>3</sup> is circumscribed on a sphere of radius 1. For such a closed surface containing a dipole it necessarily follows that the flux at the surface of the sphere of v<sub>3</sub> wrt to surface normal n is zero including at the points where the surface of sphere touches the cube walls. At the finite time singularity on the sphere a rotation boundary condition is deduced. It is shown that v<sub>3</sub> is spatially finite on the Riemann Sphere and the forcing is oscillatory in y<sub>3</sub> component if the velocity v3</sub> is. It is true that . A boundary condition on the sphere shows the rotation of a sphere of viscous fluid. Finally on the sphere a solution for v3</sub> is obtained which is proven to be Hölder continuous and it is shown that it is possible to extend Hölder continuity on the sphere uniquely to all of the interior of the ball.展开更多
We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation(■t-1/2△)u=u◇Win d-dimensional space, where W is a mean zero Gaussian no...We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation(■t-1/2△)u=u◇Win d-dimensional space, where W is a mean zero Gaussian noise with temporal covariance γo and spatial covariance given by a spectral density μ(ζ).We assume that γo(t)≤c|t|^-α0 and |μ(ζ)|≤c ∏di=1|ζi|^-αi or|μ(ζ)|≤c|ζ|^-α, where α,α1,…,αd can take both positive and negative values.展开更多
The Hǒlder continuity is proved for the gradient of the solution Jo the one-sided obstacle problem of the following variational inequality in the case 1<p<2
The Hoider continuity is proved to bounded solutions of degenerate elliptic e-quations involving measures. The structural conditions of the equation are more general and therestrictions on the structural coofficients ...The Hoider continuity is proved to bounded solutions of degenerate elliptic e-quations involving measures. The structural conditions of the equation are more general and therestrictions on the structural coofficients are weaker.展开更多
文摘This article gives a general model using specific periodic special functions, which is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of cells of the 3-Torus. Satisfying a divergence-free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which have finite time singularities can occur starting with the first derivative and higher with respect to time. The existence of a subspace of the solution space where v<sub>3</sub> is continuous and {C, y<sub>1</sub>, y<sub>1</sub><sup>2</sup>}, is linearly independent in the additive argument of the solution in terms of the Lambert W function, (y<sub>1</sub><sup>2</sup>=y<sub>2</sub>, C∈R) together with the condition v<sub>2</sub>=-2y<sub>1</sub>v<sub>1</sub>. On this subspace, the Biot Savart Law holds exactly [see Section 2 (Equation (13))]. Also on this subspace, an expression X (part of PNS equations) vanishes which contains all the expressions in derivatives of v<sub>1</sub> and v<sub>2</sub> and the forcing terms in the plane which are related as with the cancellation of all such terms in governing PDE. The y<sub>3</sub> component forcing term is arbitrarily small in ε ball where Weierstrass P functions touch the center of the ball both for inviscid and viscous cases. As a result, a significant simplification occurs with a v<sub>3 </sub>only governing PDE resulting. With viscosity present as v changes from zero to the fully viscous case at v =1 the solution for v<sub>3</sub> reaches a peak in the third component y<sub>3</sub>. Consequently, there exists a dipole which is not centered at the center of the cell of the Lattice. Hence since the dipole by definition has an equal in magnitude positive and negative peak in y<sub>3</sub>, then the dipole Riemann cut-off surface is covered by a closed surface which is the sphere and where a given cell of dimensions [-1, 1]<sup>3</sup> is circumscribed on a sphere of radius 1. For such a closed surface containing a dipole it necessarily follows that the flux at the surface of the sphere of v<sub>3</sub> wrt to surface normal n is zero including at the points where the surface of sphere touches the cube walls. At the finite time singularity on the sphere a rotation boundary condition is deduced. It is shown that v<sub>3</sub> is spatially finite on the Riemann Sphere and the forcing is oscillatory in y<sub>3</sub> component if the velocity v3</sub> is. It is true that . A boundary condition on the sphere shows the rotation of a sphere of viscous fluid. Finally on the sphere a solution for v3</sub> is obtained which is proven to be Hölder continuous and it is shown that it is possible to extend Hölder continuity on the sphere uniquely to all of the interior of the ball.
基金supported by an NSERC granta startup fund of University of Albertasupported by Martin Hairer’s Leverhulme Trust leadership award
文摘We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation(■t-1/2△)u=u◇Win d-dimensional space, where W is a mean zero Gaussian noise with temporal covariance γo and spatial covariance given by a spectral density μ(ζ).We assume that γo(t)≤c|t|^-α0 and |μ(ζ)|≤c ∏di=1|ζi|^-αi or|μ(ζ)|≤c|ζ|^-α, where α,α1,…,αd can take both positive and negative values.
基金in part by Zhongshan University Science Research Fund
文摘The Hǒlder continuity is proved for the gradient of the solution Jo the one-sided obstacle problem of the following variational inequality in the case 1<p<2
文摘The Hoider continuity is proved to bounded solutions of degenerate elliptic e-quations involving measures. The structural conditions of the equation are more general and therestrictions on the structural coofficients are weaker.
基金Supported by the Science Research Foundations for the Doctoral Program of Guilin University of Electronic Technology under Grant(UF09007Y)the Guangxi Natural Science Foundations under Grant(2010GXNSB013049)the National Natural Science Foundations under Grant(11101100,71001015)
