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Register a new userRegularity of Solutions of the Camassa-Holm Equations with Fractional Laplacian Viscosity
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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.
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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).
It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.
In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.
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}).
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,mu_X)$ satisfying a $2$-Poincare inequality. Given a bounded domain $Omegasubset X$ with $mu_X(XsetminusOmega)>0$, and a function $f$ in the Besov class $B^theta_{2,2}(X)cap L^2(X)$, we study the problem of finding a function $uin B^theta_{2,2}(X)$ such that $u=f$ in $XsetminusOmega$ and $mathcal{E}_theta(u,u)le mathcal{E}_theta(h,h)$ whenever $hin B^theta_{2,2}(X)$ with $h=f$ in $XsetminusOmega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Holder continuous on $Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
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