44 research outputs found
Convergence of the regularized short pulse equation to the short pulse one
We consider the regularized short-pulse equation, which contains nonlinear
dis- persive effects. We prove that as the diffusion parameter tends to zero,
the solutions of the dispersive equation converge to discontinuous weak
solutions of the short-pulse one. The proof relies on deriving suitable a
priori estimates together with an application of the compensated compactness
method in the Lp setting
Wellposedness results for the short pulse equation
The short pulse equation provides a model for the propagation of ultra-short
light pulses in silica optical fibers. It is a nonlinear evolution equation. In
this paper the wellposedness of bounded solutions for the homogeneous initial
boundary value problem and the Cauchy problem associated to this equation are
studied.Comment: arXiv admin note: text overlap with arXiv:1310.701
Oleinik type estimates for the Ostrovsky-Hunter eequation
The Ostrovsky-Hunter equation provides a model for small-amplitude long waves
in a rotating fluid of finite depth. It is a nonlinear evolution equation. In
this paper we study the well-posedness for the Cauchy problem associated to
this equation within a class of bounded discontinuous solutions. We show that
we can replace the Kruzkov-type entropy inequalities by an Oleinik-type
estimate and prove uniqueness via a nonlocal adjoint problem. An implication is
that a shock wave in an entropy weak solution to the Ostrovsky-Hunter equation
is admissible only if it jumps down in value (like the inviscid Burgers
equation)
Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one
We consider the Ostrovsky equation, which contains nonlinear dispersive
effects. We prove that as the diffusion parameter tend to zero, the solutions
of the dispersive equation converge to discontinuous weak solutions of the
Ostrovsky-Hunter equation. The proof relies on deriving suitable a priori
estimates together with an application of the compensated compactness method in
the L^p setting
Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects
AbstractThe Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation
Well-posedness result for the Kuramoto–Velarde equation
AbstractThe Kuramoto–Velarde equation describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface in a microgravity environment. Under appropriate assumption on the initial data, of the time T, and the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation
On the wellposedness of the exp-Rabelo equation
The exp-Rabelo equation describes pseudo-spherical surfaces. It is a
nonlinear evolution equation. In this paper the wellposedness of bounded from
above solutions for the initial value problem associated to this equation is
studied
H4-Solutions for the Olver–Benney equation
AbstractThe Olver–Benney equation is a nonlinear fifth-order equation, which describes the interaction effects between short and long waves. In this paper, we prove the global existence of solutions of the Cauchy problem associated with this equation
Convergence of the Solutions on the Generalized Korteweg–de Vries Equation*
We consider the generalized Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp setting