In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a ...In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.展开更多
We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing s...We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.展开更多
In this paper we deal with the interaction of three conormal waves for a class of third-order nonlinear strictly hyperbolic equations, in which two conormal waves are tangent. By the same argument, we may also discuss...In this paper we deal with the interaction of three conormal waves for a class of third-order nonlinear strictly hyperbolic equations, in which two conormal waves are tangent. By the same argument, we may also discuss the similar problem for equation system of compressible fluid flow and obtain similar conclusions.展开更多
The interaction of three conormal waves for semi-linear strictly hyperbolic equations of third order is considered. Let Σi, i = 1, 2, 3, be smooth characteristic surfaces for P= Da(D_t￣2 -△) intersecting transversa...The interaction of three conormal waves for semi-linear strictly hyperbolic equations of third order is considered. Let Σi, i = 1, 2, 3, be smooth characteristic surfaces for P= Da(D_t￣2 -△) intersecting transversally at the origin. Suppose that the solution u to Pu = f(t, x ,y)D u), ≤2 is conormal to Σi, i = 1, 2, 3, for t < 0. The author uses Bony's second microlocajization techniques and commutator arguments to conclude that the new singularities a short time after the triple interaction lie on the surface of the light cone Γ over the origin plus the surfaces obtained by flow-outs of the lines of intersection Γ ∩ Σi and Σi∩ Σj, i, j = 1, 2, 3.展开更多
This paper is devoted to study Cauchy problems of multidimensional semilinear strictly hyperbolic equations of second order with strongly singular initial data,where the derivatives of the initial data have discontinu...This paper is devoted to study Cauchy problems of multidimensional semilinear strictly hyperbolic equations of second order with strongly singular initial data,where the derivatives of the initial data have discontinuity on two smooth curves transversally intersecting each other.The existence of the solution is proved,meanwhile,it is precisely discribed the flowery structure of the singularity of the solution.展开更多
The author studies the technique of paradifferential operator defined on a space of conormaldistribution with three indeces,and then use this technique to prove that a progressing wavewhich hits the boundary is reflec...The author studies the technique of paradifferential operator defined on a space of conormaldistribution with three indeces,and then use this technique to prove that a progressing wavewhich hits the boundary is reflected according to the usual law.展开更多
This paper studies the initial boundary problems for semilinear strictly hyperbolic second-order equations with discontinuous data. Under the uniform Lopatinski boundary condition, the local existence theorem of disco...This paper studies the initial boundary problems for semilinear strictly hyperbolic second-order equations with discontinuous data. Under the uniform Lopatinski boundary condition, the local existence theorem of discontinuous solutions and the structure of singularities of the solutions are obtained.展开更多
We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-tr...We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.展开更多
We prove a common fixed point theorem for discontinuous,noncompatible mappings on noncomplete intuitionistic fuzzy metric spaces by using a new commutativity condition.We validate our main result by an example.
文摘In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.
文摘We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.
基金Project supported by the National Natural Science Foundation of China
文摘In this paper we deal with the interaction of three conormal waves for a class of third-order nonlinear strictly hyperbolic equations, in which two conormal waves are tangent. By the same argument, we may also discuss the similar problem for equation system of compressible fluid flow and obtain similar conclusions.
文摘The interaction of three conormal waves for semi-linear strictly hyperbolic equations of third order is considered. Let Σi, i = 1, 2, 3, be smooth characteristic surfaces for P= Da(D_t￣2 -△) intersecting transversally at the origin. Suppose that the solution u to Pu = f(t, x ,y)D u), ≤2 is conormal to Σi, i = 1, 2, 3, for t < 0. The author uses Bony's second microlocajization techniques and commutator arguments to conclude that the new singularities a short time after the triple interaction lie on the surface of the light cone Γ over the origin plus the surfaces obtained by flow-outs of the lines of intersection Γ ∩ Σi and Σi∩ Σj, i, j = 1, 2, 3.
文摘This paper is devoted to study Cauchy problems of multidimensional semilinear strictly hyperbolic equations of second order with strongly singular initial data,where the derivatives of the initial data have discontinuity on two smooth curves transversally intersecting each other.The existence of the solution is proved,meanwhile,it is precisely discribed the flowery structure of the singularity of the solution.
文摘The author studies the technique of paradifferential operator defined on a space of conormaldistribution with three indeces,and then use this technique to prove that a progressing wavewhich hits the boundary is reflected according to the usual law.
基金Research supported by the Natural Science Foundation of Fujian Province, China.
文摘This paper studies the initial boundary problems for semilinear strictly hyperbolic second-order equations with discontinuous data. Under the uniform Lopatinski boundary condition, the local existence theorem of discontinuous solutions and the structure of singularities of the solutions are obtained.
基金I+D MEC Projects No.MTM 2005-02541,MTM 2004-03882Junta de Andalucfa Grants FQM 0199,FQM 0194,FQM 1215the PCI Project No.A/4044/05 of the Spanish AECI
文摘We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.
文摘We prove a common fixed point theorem for discontinuous,noncompatible mappings on noncomplete intuitionistic fuzzy metric spaces by using a new commutativity condition.We validate our main result by an example.
