A generalization of the usual Green function to a kind of nonlinear elliptic equation of divergence form is discussed. The regularity and comparison principle of Green function in the sense of distribution are shown.
For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H r...For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H rmander. As an application of our result, we show that the solution of three- dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.展开更多
For a class of special three-dimensional quasilinear wave equations, we study the blowup mechanism of classical solutions. More precisely, under the nondegenerate conditions, any radially symmetric solution with small...For a class of special three-dimensional quasilinear wave equations, we study the blowup mechanism of classical solutions. More precisely, under the nondegenerate conditions, any radially symmetric solution with small initial data is shown to develop singularities in the second order derivaties while the first order derivatives and itself remain continuous, moreover the blowup of solution is of “cusp type”.展开更多
For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove that the first-order derivatives of ρ, υ blow-up at the blow-up time...For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove that the first-order derivatives of ρ, υ blow-up at the blow-up time, while ρ, υ remain continuous. In particular, in the irrotational case we prove S. Alinhac’s statement.展开更多
For a special class of quasilinear wave equations with small initial data which satisfy the nondegenerate assumption, the authors prove that the radially symmetric solution develops singularities in the second order d...For a special class of quasilinear wave equations with small initial data which satisfy the nondegenerate assumption, the authors prove that the radially symmetric solution develops singularities in the second order derivatives in finite time while the first order derivatives and the solution itself remain continuous and small. More precisely, it turns out that this solution is a "geometric blowup solution of cusp type", according to the terminology posed by S. Alinhac[2].展开更多
For 2-D quasilinear wave equations with cubic nonlinearity and small initial data, we not only show that the solutions blow up in finite time but also give a complete asymptotic expansion of the lifespan of classical ...For 2-D quasilinear wave equations with cubic nonlinearity and small initial data, we not only show that the solutions blow up in finite time but also give a complete asymptotic expansion of the lifespan of classical solutions. Hence we solve a problem posed by S. Alinhac and A. Hoshiga. Moreover, as an application of this result, we prove the blowup of solutions for the nonlinear vibrating membrane equations.展开更多
基金Supported by Beijing Jiaotong University Science Research Foundation (2004SM056)
文摘A generalization of the usual Green function to a kind of nonlinear elliptic equation of divergence form is discussed. The regularity and comparison principle of Green function in the sense of distribution are shown.
基金Project supported by the Zheng Ge Ru FoundationTianyuan Foundation of China
文摘For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H rmander. As an application of our result, we show that the solution of three- dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.
文摘For a class of special three-dimensional quasilinear wave equations, we study the blowup mechanism of classical solutions. More precisely, under the nondegenerate conditions, any radially symmetric solution with small initial data is shown to develop singularities in the second order derivaties while the first order derivatives and itself remain continuous, moreover the blowup of solution is of “cusp type”.
基金Project supported by the Tianyuan Foundation of ChinaLab. of Math, for Nonlinear Problems. Fudan. Univ.
文摘For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove that the first-order derivatives of ρ, υ blow-up at the blow-up time, while ρ, υ remain continuous. In particular, in the irrotational case we prove S. Alinhac’s statement.
文摘For a special class of quasilinear wave equations with small initial data which satisfy the nondegenerate assumption, the authors prove that the radially symmetric solution develops singularities in the second order derivatives in finite time while the first order derivatives and the solution itself remain continuous and small. More precisely, it turns out that this solution is a "geometric blowup solution of cusp type", according to the terminology posed by S. Alinhac[2].
基金I thank Prof. S.Alinhac and Prof. Xin Zhouping very much for giving me the invaluable discussion and guidance. This work was supported by the Tianyuan Foundation of China.
文摘For 2-D quasilinear wave equations with cubic nonlinearity and small initial data, we not only show that the solutions blow up in finite time but also give a complete asymptotic expansion of the lifespan of classical solutions. Hence we solve a problem posed by S. Alinhac and A. Hoshiga. Moreover, as an application of this result, we prove the blowup of solutions for the nonlinear vibrating membrane equations.
