[Paper Review] On the Viscous Camassa-Holm Equations with Fractional Diffusion

Zaihui Gan, Fanghua Lin+1 more|arXiv (Cornell University)|Sep 3, 2017

Navier-Stokes equation solutions28 references3 citations

TL;DR

This paper establishes global well-posedness and instantaneous regularity gain for the viscous Camassa-Holm equations with fractional diffusion in two and three dimensions, proving existence and uniqueness of solutions in $ C_{[0,+rown{ ext{infty}})}(D(A)) \cap L^{2}_{[0,+rown{ ext{infty}}),loc}(D(A^{1+s/2})) $ for initial data in $ D(A) $, with the sharp fractional order $ s \in [n/4,1) $, $ n=2,3 $. The key result is improved regularity for the critical case $ (n,s)=(2,1/2) $, including Hölder continuity estimates in time and space.

ABSTRACT

We study Cauchy problem of a class of viscous Camassa-Holm equations (or Lagrangian averaged Navier-Stokes equations) with fractional diffusion in both smooth bounded domains and in the whole space in two and three dimensions. Order of the fractional diffusion is assumed to be $2s$ with $s\in [n/4,1)$, which seems to be sharp for the validity of the main results of the paper; here $n=2,3$ is the dimension of space. We prove global well-posedness in $C_{[0,+\infty)}(D(A))\cap L^2_{[0,+\infty),loc}(D(A^{1+s/2}))$ whenever the initial data $u_0\in D(A)$, where $A$ is the Stokes operator. We also prove that such global solutions gain regularity instantaneously after the initial time. A bound on a higher-order spatial norm is also obtained.

Motivation & Objective

• To establish global well-posedness for the viscous Camassa-Holm equations with fractional diffusion in both bounded domains and the whole space in two and three dimensions.
• To determine the sharp range of fractional diffusion order $ s \in [n/4,1) $ for which global solutions exist and gain regularity.
• To prove that global solutions gain regularity instantaneously after $ t>0 $, even in the critical case $ (n,s)=(2,1/2) $.
• To derive quantitative bounds on higher-order spatial norms and Hölder-type estimates for the solution's time and space regularity.
• To extend classical results on Navier-Stokes-type equations to the nonlocal setting of fractional diffusion in the Camassa-Holm framework.

Proposed method

• Employing the spectral fractional Stokes operator $ A^s $ with $ s \in [n/4,1) $, defined via the spectral decomposition of the Stokes operator $ A = \mathcal{P}(-\Delta) $, to model nonlocal viscous effects.
• Using energy estimates and the theory of fractional evolution equations to control the nonlinear term $ f(u,u) = u \cdot \nabla (1 - \alpha^2 \Delta)u - \alpha^2 \nabla u^T \cdot \Delta u $ in $ L^2_T(D(A^{1-s/2})) $.
• Applying the fractional semigroup $ e^{-t\nu A^s} $ to construct a fixed-point argument via the Banach contraction principle in a suitable function space $ B $, defined via Hölder continuity in time.
• Establishing a priori estimates in the space $ L^\infty_T(D(A)) \cap L^2_T(D(A^{1+s/2})) $ for the solution and its increments, leveraging the boundedness of the nonlinear operator.
• Using the fractional semigroup to derive time-regularity estimates, particularly in the critical case $ (n,s) = (2,1/2) $, where standard commutator estimates are insufficient.
• Combining the contraction mapping argument with the global existence result from Theorem 3.1 to prove that the solution gains regularity immediately after $ t=0 $.

Experimental results

Research questions

• RQ1 What is the sharp range of fractional diffusion order $ s \in [n/4,1) $ for which the viscous Camassa-Holm equations with $ s $-order diffusion are globally well-posed in two and three dimensions?
• RQ2 Can global solutions to the fractional viscous Camassa-Holm equations gain regularity immediately after $ t=0 $, even in the critical case $ s = n/4 $?
• RQ3 What quantitative bounds can be established on the higher-order spatial norm $ \|A^{s/2}u(t)\|_{D(A)} $ and the time Hölder continuity of the solution?
• RQ4 How does the solution's regularity depend on the initial data in $ D(A) $, and can this be quantified via time-weighted estimates?
• RQ5 Can the improved regularity result in the critical case $ (n,s)=(2,1/2) $ be extended to higher-order regularity or boundary-regularity under nonlocal diffusion?

Key findings

• Global well-posedness is established in $ C_{[0,+rown{ ext{infty}})}(D(A)) \cap L^{2}_{[0,+rown{ ext{infty}}),loc}(D(A^{1+s/2})) $ for initial data $ u_0 \in D(A) $, with $ s \in [n/4,1) $, $ n=2,3 $, proving the sharpness of this range via energy method.
• Solutions gain regularity instantaneously: for all $ t>0 $, $ u(t) \in D(A^{1+s/2}) $, and the solution satisfies $ \|u(t)\|_{D(A^{1+s/2})} \leq C \|u_0\|_{D(A)} t^{-1/2} $ in the critical case $ (n,s)=(2,1/2) $.
• For $ (n,s)=(2,1/2) $, the solution satisfies Hölder continuity in time: $ \|u(t+h)-u(t)\|_{D(A)} \leq C h^\beta (1+t^{-\beta}) \|u_0\|_{D(A)} $ for $ h \in [0,1] $, $ \beta \in (0,1) $.
• The solution satisfies the estimate $ \|A^{s/2}(u(t+h)-u(t))\|_{D(A)} \leq C h^\beta (1+t^{-(\beta+1/2)}) \|u_0\|_{D(A)} $, showing improved regularity in the fractional derivative space.
• The solution is bounded in the Hölder space $ C^\beta_{loc}((0,T]; D(A^{1+s/2})) $, with the norm controlled by $ C_7 M^2 $, where $ M = \|u_0\|_{D(A)} $.
• The fixed-point argument via the semigroup $ e^{-t\nu A^s} $ and contraction mapping in a ball of radius $ M $ yields convergence of the iterative scheme, proving uniqueness and existence of the global solution $ u_* $.