Do you want to publish a course? Click here

On a quasilinear non-local Benney System

50   0   0.0 ( 0 )
 Added by Filipe Oliveira
 Publication date 2015
  fields
and research's language is English




Ask ChatGPT about the research

We study the quasilinear non-local Benney System $$left{begin{array}{llll} iu_t+u_{xx}=|u|^2u+buv v_t+a(int_{mathbf{R}^+}v^2dx)v_x=-b(|u|^2)_x,quad (x,t)inmathbf{R}^+times [0,T],, T>0. end{array}right.$$ We establish the existence and uniqueness of strong local solutions to the corresponding Cauchy problem and show, under certain conditions, the blow-up of such solutions in finite time. Furthermore, we prove the existence of global weak solutions and exhibit bound-state solutions to this system.



rate research

Read More

Proving local well-posedness for quasilinear problems in pdes presents a number of difficulties, some of which are universal and others of which are more problem specific. While a common standard, going back to Hadamard, has existed for a long time, there are by now both many variations and many misconceptions in the subject. The aim of these notes is to collect a number of both classical and more recent ideas in this direction, and to assemble them into a cohesive road map that can be then adapted to the readers problem of choice.
We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ left{begin{array}{rlll} e^{-itheta}u_t&=&u_{xx}-|u|^2u-alpha g(v)u& v_t+(f(v))_x&=&alpha (g(v)|u|^2)_x& end{array}right. qquad xinmathbb{R},, t geq 0, $$ which can describe the interaction between a laser beam and a fluid flow (see [Aranson, Kramer, Rev. Med. Phys. 74 (2002)]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore we prove the existence of standing waves of the form $(u(t,x),v(t,x))=(U(x),V(x))$ in several cases.
99 - Janna Lierl 2017
We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
77 - Chengyang Shao 2020
The paper studies the long time behavior of a system that describes the motion of a piece of elastic membrane driven by surface tension and inner air pressure. The system is a degenerate quasilinear hyperbolic one that involves the mean curvature, and also includes a damping term that models the dissipative nature of genuine physical systems. With the presence of damping, a small perturbation of the sphere converges exponentially in time to the sphere, and without the damping the evolution that is $varepsilon$-close to the sphere has life span longer than $varepsilon^{-1/6}$. Both results are proved using an improved Nash-Moser-Hormander theorem.
This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a semi-Markov modulated GBM model the locally risk minimizing price function satisfies a special case of this problem. We study the well-posedness of the problem via a Volterra integral equation of second kind. A probabilistic approach, in particular the method of conditioning on stopping times is used for showing uniqueness.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا