In this paper we answer all the questions about the conjectures of Glimm and Lax on genericproperties of solutions. We prove that the discotinuous points of almost every solution with L∞,bounded varistion or contrinu...In this paper we answer all the questions about the conjectures of Glimm and Lax on genericproperties of solutions. We prove that the discotinuous points of almost every solution with L∞,bounded varistion or contrinuous data are dense in the upper half-plane minus the closure of the setof central simple waves. It is also proved that if the equation is analytic,then the solutions withpiecewise analytic data are piecewise analytic,and the shock curves are also piecewise analytic. Wedisprove the conjecture which claims that almost every solution with C^k data is 'bad' enough, and provethat every solution with C^k data possesses nice propetied, i.e. when k≥4 the generic property ofsolutions is piecewise C^k,and hence is 'good' enough.For the proof of the generic property withC^k (k≥4) data, the idea of transversality in the theory of singular points is essential.展开更多
In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operat...In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operators, the existence, uniqueness, regularity properties of solutions are established. By choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.展开更多
In this paper we show the decay of solutions to the initial-boundary value problem for some nonlinear hyperbolic equation with a nonlinear dissipative term, by using a difference inequality.
文摘In this paper we answer all the questions about the conjectures of Glimm and Lax on genericproperties of solutions. We prove that the discotinuous points of almost every solution with L∞,bounded varistion or contrinuous data are dense in the upper half-plane minus the closure of the setof central simple waves. It is also proved that if the equation is analytic,then the solutions withpiecewise analytic data are piecewise analytic,and the shock curves are also piecewise analytic. Wedisprove the conjecture which claims that almost every solution with C^k data is 'bad' enough, and provethat every solution with C^k data possesses nice propetied, i.e. when k≥4 the generic property ofsolutions is piecewise C^k,and hence is 'good' enough.For the proof of the generic property withC^k (k≥4) data, the idea of transversality in the theory of singular points is essential.
文摘In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operators, the existence, uniqueness, regularity properties of solutions are established. By choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.
文摘In this paper we show the decay of solutions to the initial-boundary value problem for some nonlinear hyperbolic equation with a nonlinear dissipative term, by using a difference inequality.
