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1.
We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (u
t+uux)x=1/2u
x
2
with the simplest initial data such that u
x blows up in finite time. We prove that the zero-viscosity limit selects a unique global weak solution of the partial differential equation without viscosity. We also present numerical experiments which indicate that the zero-dispersion limit selects a different global weak solution of the same initial-value problem. 相似文献
2.
We establish here the global existence and uniqueness of admissible (both dissipative and conservative) weak solutions to
a canonical asymptotic equation () for weakly nonlinear solutions of a class of nonlinear variational wave equations with any L
2(ℝ) initial datum. We use the method of Young measures and mollification techniques.
Accepted April 25, 2000?Published online November 16, 2000 相似文献
3.
We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R
d
, by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions. 相似文献
4.
Asymptotic Variational Wave Equations 总被引:1,自引:0,他引:1
Alberto Bressan Ping Zhang Yuxi Zheng 《Archive for Rational Mechanics and Analysis》2007,183(1):163-185
We investigate the equation (u
t
+(f(u))
x
)
x
=f
′ ′(u) (u
x
)2/2 where f(u) is a given smooth function. Typically f(u)=u
2/2 or u
3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u
tt
− c(u) (c(u)u
x
)
x
=0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data. 相似文献
5.
Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation
x+f(t,x,x′,x″)=0