In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al...In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al.(2008).This family of solutions has finite energy in any bounded domain of R3,but unbounded energy in R3.Based on similarity coordinates,energy estimates and the Nash-Moser-H?rmander iteration scheme,we show that these solutions are asymptotically stable in the backward light-cone of the singularity.Furthermore,the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data.Finally,the result also shows that in the absence of physical boundaries,the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.展开更多
The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the a...The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the analytical blowup solutions of the N-dimensional （N ≥ 2） Navier-Stokes equations, we extend the similar structure to construct an analytical family of solutions for the pressureless Navier-Stokes equations with a normal viscosity term （μ（ρ）｜ u｜^α u）.展开更多
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|^(p-2)?u) =-|u|^(β-1) u + α|u|^(q-2 )u,where p > 1, β > 0, q≥1 and α > 0. By using Gagliard...In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|^(p-2)?u) =-|u|^(β-1) u + α|u|^(q-2 )u,where p > 1, β > 0, q≥1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.12231016 and 12071391)Guangdong Basic and Applied Basic Research Foundation(Grant No.2022A1515010860)。
文摘In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al.(2008).This family of solutions has finite energy in any bounded domain of R3,but unbounded energy in R3.Based on similarity coordinates,energy estimates and the Nash-Moser-H?rmander iteration scheme,we show that these solutions are asymptotically stable in the backward light-cone of the singularity.Furthermore,the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data.Finally,the result also shows that in the absence of physical boundaries,the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.
基金Supported by the NSFC of China (1087117510931007+1 种基金10901137)supported by the Scientific Research Fund of Education Department of Zhejiang Province (Y200803203)
文摘The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the analytical blowup solutions of the N-dimensional （N ≥ 2） Navier-Stokes equations, we extend the similar structure to construct an analytical family of solutions for the pressureless Navier-Stokes equations with a normal viscosity term （μ（ρ）｜ u｜^α u）.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371286 and 11401458)the Special Fund of Education Department (Grant No. 2013JK0586)the Youth Natural Science Grant of Shaanxi Province of China (Grant No. 2013JQ1015)
文摘In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|^(p-2)?u) =-|u|^(β-1) u + α|u|^(q-2 )u,where p > 1, β > 0, q≥1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
