Yuwen Luo | University of British Columbia (original) (raw)

Papers by Yuwen Luo

Research paper thumbnail of Regularity criteria in weak L for 3D incompressible Navier-Stokes equations

Research paper thumbnail of Regularity Criteria in Weak <i>L</i><sup>3</sup> for 3D Incompressible Navier-Stokes Equations

Funkcialaj Ekvacioj, 2015

We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Na... more We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Navier-Stokes equations. Let B r denote concentric balls in R 3 with radius r. We will show that if p ∈ L m (0, 1; L 1 (B 2)), m > 2, and if u is sufficiently small in L ∞ (0, 1; L 3,∞ (B 2)), without any assumption on its gradient, then u is bounded in B 1 × (1 10 , 1). It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.

Research paper thumbnail of Regularity of weak solutions to the magneto-hydrodynamics equations in terms of the direction of velocity

In this article, we study the regularity of weak solutions to the 3D incompressible magneto-hydro... more In this article, we study the regularity of weak solutions to the 3D incompressible magneto-hydrodynamics equations. We obtain a new class of regularity criteria in terms of the direction of the velocity. Our result extend some results known for incompressible Navier-Stokes equations.

Research paper thumbnail of Well-Posedness Of A Cauchy Problem Involving Nonlinear Fractal Dissipative Equations

This paper studies the well-posedness of fractal dissipative equations with the initial data u0 i... more This paper studies the well-posedness of fractal dissipative equations with the initial data u0 in Lebesgue spaces. For u0 ∈ L r (R n), r > n/(2α − 1) and α > 1/2 we prove that the fractal dissipative equation has a unique local solution. For the critical case r = n/(2α − 1), the equation has a unique local solution in C([0, T ]; L r (R n)) ∩ C([0, T ]; L p (R n)) for each T > 0. Furthermore, if the L r norm of the initial data is small enough, then the solution is global.

Research paper thumbnail of On the regularity of generalized MHD equations

Journal of Mathematical Analysis and Applications, 2010

Research paper thumbnail of ftp ejde.math.txstate.edu REGULARITY OF WEAK SOLUTIONS TO THE MAGNETO-HYDRODYNAMICS EQUATIONS IN TERMS OF THE DIRECTION OF VELOCITY

Abstract. In this article, we study the regularity of weak solutions to the 3D incompressible mag... more Abstract. In this article, we study the regularity of weak solutions to the 3D incompressible magneto-hydrodynamics equations. We obtain a new class of regularity criteria in terms of the direction of the velocity. Our result extend some results known for incompressible Navier-Stokes equations.

Research paper thumbnail of ftp ejde.math.txstate.edu REGULARITY OF GENERALIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VELOCITY

Abstract. In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS... more Abstract. In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms (−∆) α u. It is proved that if div(u/|u|) ∈ L p (0, T; L q (R 3)) with

Research paper thumbnail of Partial Regularity of Generalized MHD Equations

In this paper,the regularity of 3D generalized magneto-hydrodynamics(GMHD) equations with fractio... more In this paper,the regularity of 3D generalized magneto-hydrodynamics(GMHD) equations with fractional dissipative terms(-Δ) αand(-Δ) β is studied. The equations contain the well-known Navier-Stokes equations and magneto-hydrodynamics equations. The regularity problem of Navier-Stokes equation and generalized type,such as the magneto-hydrodynamics equation and the generalized Navier-Stokes equations,were studied extensively. But the problem is still unsettled now. Some researchers turned to study the regularities criterion of Naveir-Stokes equations in terms of the components of velocity,and got some of useful results. Since MHD and GMHD are far more difficult than Navier-Stokes equations,there are few similar results of these two equations. Using energy integral method,the regularity of GMHD equations in terms of two components of velocity is studied,and the results does not depend on the magnetic field. The special case α = β is considered in the paper. Let u =(u1,u2,u3) , =(u1,u2,0) ,0 α = β 2/3,the initial velocity and magnetic field satisfied u0,b0∈H1(R3) . Under these conditions,it is proved that if ∈LP(0,T;Lq(R3) ) with(2α) /p+3/q ≤2α on [0,T],or if ∈L2((2α) /(2α-r))(0,T;(R3)) with 0≤r≤α,the solution remains smooth on[0,T].

Research paper thumbnail of Regularity of generalized Navier-Stokes equations in terms of direction of the velocity

In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equation... more In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms (−∆)αu. It is proved that if div(u/|u|) ∈ Lp(0, T ; Lq(R3)) with 2α p + 3 q ≤ 2α− 3 2 , 6 4α− 3 < q ≤ ∞. then any smooth on GNS in [0, T ) remains smooth on [0, T ].

Research paper thumbnail of Regularity criteria in weak L for 3D incompressible Navier-Stokes equations

Research paper thumbnail of Regularity Criteria in Weak <i>L</i><sup>3</sup> for 3D Incompressible Navier-Stokes Equations

Funkcialaj Ekvacioj, 2015

We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Na... more We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Navier-Stokes equations. Let B r denote concentric balls in R 3 with radius r. We will show that if p ∈ L m (0, 1; L 1 (B 2)), m > 2, and if u is sufficiently small in L ∞ (0, 1; L 3,∞ (B 2)), without any assumption on its gradient, then u is bounded in B 1 × (1 10 , 1). It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.

Research paper thumbnail of Regularity of weak solutions to the magneto-hydrodynamics equations in terms of the direction of velocity

In this article, we study the regularity of weak solutions to the 3D incompressible magneto-hydro... more In this article, we study the regularity of weak solutions to the 3D incompressible magneto-hydrodynamics equations. We obtain a new class of regularity criteria in terms of the direction of the velocity. Our result extend some results known for incompressible Navier-Stokes equations.

Research paper thumbnail of Well-Posedness Of A Cauchy Problem Involving Nonlinear Fractal Dissipative Equations

This paper studies the well-posedness of fractal dissipative equations with the initial data u0 i... more This paper studies the well-posedness of fractal dissipative equations with the initial data u0 in Lebesgue spaces. For u0 ∈ L r (R n), r > n/(2α − 1) and α > 1/2 we prove that the fractal dissipative equation has a unique local solution. For the critical case r = n/(2α − 1), the equation has a unique local solution in C([0, T ]; L r (R n)) ∩ C([0, T ]; L p (R n)) for each T > 0. Furthermore, if the L r norm of the initial data is small enough, then the solution is global.

Research paper thumbnail of On the regularity of generalized MHD equations

Journal of Mathematical Analysis and Applications, 2010

Research paper thumbnail of ftp ejde.math.txstate.edu REGULARITY OF WEAK SOLUTIONS TO THE MAGNETO-HYDRODYNAMICS EQUATIONS IN TERMS OF THE DIRECTION OF VELOCITY

Abstract. In this article, we study the regularity of weak solutions to the 3D incompressible mag... more Abstract. In this article, we study the regularity of weak solutions to the 3D incompressible magneto-hydrodynamics equations. We obtain a new class of regularity criteria in terms of the direction of the velocity. Our result extend some results known for incompressible Navier-Stokes equations.

Research paper thumbnail of ftp ejde.math.txstate.edu REGULARITY OF GENERALIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VELOCITY

Abstract. In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS... more Abstract. In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms (−∆) α u. It is proved that if div(u/|u|) ∈ L p (0, T; L q (R 3)) with

Research paper thumbnail of Partial Regularity of Generalized MHD Equations

In this paper,the regularity of 3D generalized magneto-hydrodynamics(GMHD) equations with fractio... more In this paper,the regularity of 3D generalized magneto-hydrodynamics(GMHD) equations with fractional dissipative terms(-Δ) αand(-Δ) β is studied. The equations contain the well-known Navier-Stokes equations and magneto-hydrodynamics equations. The regularity problem of Navier-Stokes equation and generalized type,such as the magneto-hydrodynamics equation and the generalized Navier-Stokes equations,were studied extensively. But the problem is still unsettled now. Some researchers turned to study the regularities criterion of Naveir-Stokes equations in terms of the components of velocity,and got some of useful results. Since MHD and GMHD are far more difficult than Navier-Stokes equations,there are few similar results of these two equations. Using energy integral method,the regularity of GMHD equations in terms of two components of velocity is studied,and the results does not depend on the magnetic field. The special case α = β is considered in the paper. Let u =(u1,u2,u3) , =(u1,u2,0) ,0 α = β 2/3,the initial velocity and magnetic field satisfied u0,b0∈H1(R3) . Under these conditions,it is proved that if ∈LP(0,T;Lq(R3) ) with(2α) /p+3/q ≤2α on [0,T],or if ∈L2((2α) /(2α-r))(0,T;(R3)) with 0≤r≤α,the solution remains smooth on[0,T].

Research paper thumbnail of Regularity of generalized Navier-Stokes equations in terms of direction of the velocity

In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equation... more In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms (−∆)αu. It is proved that if div(u/|u|) ∈ Lp(0, T ; Lq(R3)) with 2α p + 3 q ≤ 2α− 3 2 , 6 4α− 3 < q ≤ ∞. then any smooth on GNS in [0, T ) remains smooth on [0, T ].