The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Gre...The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Green's function and iteration method, we improve the L2-decay rate in Tan and Wang(2013)and Tan and Wu(2012)when(ρ0-ˉρ,m)˙B-s1,∞×˙B-s+11,∞with s∈[0,2]is bounded.In particular,it holds that the density converges to its equilibrium state at the rate(1+t)-34-s2 in L2-norm and the momentum decays at the rate(1+t)-54-s2 in L2-norm.Moreover,under a weaker and more general condition on the initial data,we show that the density and the momentum have different pointwise estimates in dimension d with d 3on both space variable x and time variable t as|Dαx(ρ-ˉρ)|C(1+t)-d2-|α|2(1+|x|21+t)-rwith r>d2and|Dαxm|C(1+t)-d2-|α|+12(1+|x|21+t)-d2 by a more elaborate analysis on the Green’s function.These results improve those in Wang and Yang(2001),where the density and the velocity(the momentum)have the same pointwise estimates.展开更多
In the present study, the flow visualizations were performed around the NACA 0012 models which differ in aspect ratios. We discussed the effects of the aspect ratio in the test models. Additionally the unsteady, two-d...In the present study, the flow visualizations were performed around the NACA 0012 models which differ in aspect ratios. We discussed the effects of the aspect ratio in the test models. Additionally the unsteady, two-dimensional, compressible Euler equations were solved for the NACA 0012 airfoil. Experiments were performed utilizing the conventional gas driven shock tube as the intermittent transonic wind tunnel. The aspect ratios of the models are about 0.86 and 1.5, respectively. The Mach numbers M 2 are about 0.84. The Reynolds numbers of the present experimental conditions were constant that Re based on chord length is about 4.0×10 5 . The results are as follows: in different aspect ratios, the difference of the shock wave location is confirmed though the Mach number and Reynolds number are same. It indicates the different correction Mach number by the effects of the side wall boundary layer though the nominal Mach number measured the same value. Also, on the difference of shock wave location for the effects of the aspect ratio, the tend of CFD shows the qualitative agreement with the result of an experiment.展开更多
In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field...In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy's law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary efects and a vacuum.展开更多
The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singu...The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system.On the one hand,the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrdinger equation.On the other hand,in the limit system,it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.展开更多
基金Supportedin part by National Natural Science Foundation of China (No.10661007)Natural Science Foundation of Jiangxi Province (No .2007GZS0811)Research Foundation of East China Jiaotong University
基金Supported by the National Natural Science Foundation of China(11501525)the Outstanding Youth Foundation of Science and Technology Innovation of Henan Province(2018JQ0004)
基金supported by National Natural Science Foundation of China(Grant Nos.11101112 and 11231006)the Fundamental Research Funds for the Central Universities(Grant No.2232015D3-33)
文摘The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Green's function and iteration method, we improve the L2-decay rate in Tan and Wang(2013)and Tan and Wu(2012)when(ρ0-ˉρ,m)˙B-s1,∞×˙B-s+11,∞with s∈[0,2]is bounded.In particular,it holds that the density converges to its equilibrium state at the rate(1+t)-34-s2 in L2-norm and the momentum decays at the rate(1+t)-54-s2 in L2-norm.Moreover,under a weaker and more general condition on the initial data,we show that the density and the momentum have different pointwise estimates in dimension d with d 3on both space variable x and time variable t as|Dαx(ρ-ˉρ)|C(1+t)-d2-|α|2(1+|x|21+t)-rwith r>d2and|Dαxm|C(1+t)-d2-|α|+12(1+|x|21+t)-d2 by a more elaborate analysis on the Green’s function.These results improve those in Wang and Yang(2001),where the density and the velocity(the momentum)have the same pointwise estimates.
文摘In the present study, the flow visualizations were performed around the NACA 0012 models which differ in aspect ratios. We discussed the effects of the aspect ratio in the test models. Additionally the unsteady, two-dimensional, compressible Euler equations were solved for the NACA 0012 airfoil. Experiments were performed utilizing the conventional gas driven shock tube as the intermittent transonic wind tunnel. The aspect ratios of the models are about 0.86 and 1.5, respectively. The Mach numbers M 2 are about 0.84. The Reynolds numbers of the present experimental conditions were constant that Re based on chord length is about 4.0×10 5 . The results are as follows: in different aspect ratios, the difference of the shock wave location is confirmed though the Mach number and Reynolds number are same. It indicates the different correction Mach number by the effects of the side wall boundary layer though the nominal Mach number measured the same value. Also, on the difference of shock wave location for the effects of the aspect ratio, the tend of CFD shows the qualitative agreement with the result of an experiment.
基金supported by the National Natural Science Foundation of China(No.11171223)the Innovation Program of Shanghai Municipal Education Commission(No.13ZZ109)
文摘In this paper, a one-dimensional bipolar Euler-Poisson system(a hydrodynamic model) from semiconductors or plasmas with boundary efects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy's law time asymptotically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the corresponding one-dimensional compressible Euler equations with damping exponentially fast as t → +∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary efects and a vacuum.
基金Project supported by the National Natural Science Foundation of China (Nos.10801102,10771151)the Sichuan Youth Sciences and Technology Foundation (No.07ZQ026-009)the China Postdoctoral Science Foundation
文摘The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system.On the one hand,the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrdinger equation.On the other hand,in the limit system,it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.
