Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLarge Time Behavior and Convergence for the Camassa-Holm Equations with Fractional Laplacian Viscosity
79
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we consider the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity in the whole space. In stark contrast to the Camassa-Holm equations without any nonlocal effect, to our best knowledge, little has been known on the large time behavior and convergence for the nonlocal equations under study. We first study the large time behavior of solutions. We then discuss the relation between the equations under consideration and the imcompressible Navier-Stokes equations with fractional Laplacian viscosity (INSF). The main difficulty to achieve them lies in the fractional Laplacian viscosity. Fortunately, by employing some properties of fractional Laplacian, in particular, the fractional Leibniz chain rule and the fractional Gagliardo-Nirenberg-Sobolev type estimates, the high and low frequency splitting method and the Fourier splitting method, we first establish the large time behavior concerning non-uniform decay and algebraic decay of solutions to the nonlocal equations under study. In particular, under the critical case $s=dfrac{n}{4}$, the nonlocal version of Ladyzhenskayas inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the non-uniform decay and algebraic decay. On the other hand, by means of the fractional heat kernel estimates, we figure out the relation between the nonlocal equations under consideration and the equations (INSF). Specifically, we prove that the solution to the Camassa-Holm equations with nonlocal viscosity converges strongly as the filter parameter $alpharightarrow~0$ to a solution of the equations (INSF).
rate research
Read More
We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $displaystyle s=frac{n}{4}$ with $n=2,3$, we will make extra efforts for acquiring the expected estimates as obtained in the case $displaystyle frac{n}{4}<s<1$. By the aid of the fractional Leibniz rule and the nonlocal version of Ladyzhenskayas inequality, we prove the existence, uniqueness and regularity to the Camassa-Holm equations under study by the energy method and a bootstrap argument, which rely crucially on the fractional Laplacian viscosity. In particular, under the critical case $s=dfrac{n}{4}$, the nonlocal version of Ladyzhenskayas inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the desired results concernig existence, uniqueness and regularity.
In the paper, by constructing a initial data $u_{0}in B^{sigma}_{p,infty}$ with $sigma-2>max{1+frac 1 p, frac 3 2}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_{0}$ is discontinuous at $t=0$ in the norm of $B^{sigma}_{p,infty}$, which implies that the ill-posedness for the higher dimensional Camassa-Holm equations in $B^{sigma}_{p,infty}$.
We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $pin [1,infty], rin (1,infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (cite{Danchin2,Byers,HHK}).
Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given.
Series of deformed Camassa-Holm-type equations are constructed using the Lagrangian deformation and Loop algebra splittings. They are weakly integrable in the sense of modified Lax pairs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
