Interior regularity results (local) and Surface quasi-geostrophic equation: Difference between pages

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(Created page with "$ \newcommand{\R}{\mathbb{R}} $ The surface quasi-geostrophic (SQG) equation consists of an evolution equation for a scalar function $\theta: \R^+ \times \R^2 \to \R$. In the ...")
 
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Let <math>\Omega</math> be an open domain and <math> u </math> a solution of an elliptic equation in <math> \Omega </math>. The following theorems say that <math> u </math> satisfies some regularity estimates in the interior of <math> \Omega </math> (but not necessarily up the the boundary).
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The surface quasi-geostrophic (SQG) equation consists of an evolution equation for a scalar function $\theta: \R^+ \times \R^2 \to \R$. In the inviscid case the equation is $$ \theta_t + u \cdot \nabla \theta = 0,$$ where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.


== Linear equations ==
Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$


Regularity results for linear equations are applicable to nonlinear equations as well through the linearization of the equation. However, this process requires some initial regularity knowledge on the solution (since the coefficients of the linearization depend on the solution itself). Therefore, the less regularity required for the coefficients, the more useful the theorem is.
The equation is used as a toy model for the 3D [[Euler equation]] and [[Navier-Stokes]]. The main question is to determine whether the Cauchy problem is well posed. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions. The same comparison can be made between the supercritical SQG equation and [[Navier-Stokes]].


All regularity results that require some modulus of continuity or smallness condition for the coefficients rely on the idea that the solution is locally close to a solution to an equation with constant coefficients. The proof is based on an estimate on how far these two solutions are at small scales. These type of arguments are often called [[perturbation methods]].
For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.
 
From the results below for linear equations, [[De Giorgi-Nash-Moser]] and [[Krylov-Safonov]] are the only non perturbative results. Their assumptions are scale invariant in the sense that a rescaling of the solution ($u_r(x) = u(rx)$) would solve an elliptic equation with the same bounds as the original.
 
* [[De Giorgi-Nash-Moser]]
<math> {\rm div \,} A(x) Du + b(x) \cdot \nabla u = 0 </math>
 
then <math>u</math> is Holder continuous if <math>A</math> is just uniformly elliptic and <math>b</math>
is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).
 
* [[Krylov-Safonov theorem]]
<math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>
 
with <math>a_{ij}</math> unif elliptic, <math>b \in L^n</math> and <math>f \in L^n</math>, then the
solution is <math>C^\alpha</math>
 
* [[Calderon-zygmund estimate]]
<math>a_{ij}(x) u_{ij} = f</math>
 
with <math>a_{ij}</math> close enough to the identity (or continuous) and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.
 
* [[Cordes-Nirenberg estimate]]
<math>a_{ij}(x) u_{ij} = f</math>
 
with <math>a_{ij}</math> close enough to the identity uniformly and $f \in L^\infty$,
then $u$ is in $C^{1,\alpha}$
 
* [[Cordes-Nirenberg estimate improved]] (corollary of work of Caffarelli for nonlinear equations)
<math>a_{ij}(x) u_{ij} = f</math>
 
with $a_{ij}$ close enough to the identity in a scale invariant Morrey
norm in terms of $L^n$ and $f \in L^n$, then $u$ is in $C^{1,\alpha}$.
 
($a_{ij} \in VMO$ is a particular case of this)
 
*[[Schauder estimate]]
<math>a_{ij}(x) u_{ij} = f</math>
 
with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$
 
== Non linear equations ==
 
* [[De Giorgi-Nash-Moser]]
For any smooth strictly convex Lagrangian $L$, minimizers of functionals
 
$ \int_D L(\nabla u) \ dx $
 
are smooth (analytic if $L$ is analytic).
 
* [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
 
Any continuous function $u$ such that
 
$M^+(D^2 u) \geq 0 \geq M^-(D^2 u)$
 
in the viscosity sense (where $M^+$ and $M^-$ are the Pucci operators), is Holder continuous.
 
(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
 
* [[Ishii-Lions Lipschitz estimate]]
 
If $u$ solves a fully nonlinear equation
 
$F(D^2 u, Du, u, x) = 0$
 
which is degenerate elliptic but satisfies some structure conditions and some smoothness assumptions respect to $x$, then $u$ is Lipschitz.
 
(The proof of this is based on the uniqueness technique for viscosity solutions)
 
* [[Lin theorem]]
 
Any continuous function $u$ such that
 
$0 \geq M^-(D^2 u)$
 
in the viscosity sense, is twice differentiable almost everywhere and $D^2 u \in L^\varepsilon$.
 
(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
 
* [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
If $u$ solves a fully nonlinear equation
 
$F(D^2 u, x) = 0$
 
which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.
 
* [[Evans-Krylov theorem]]
If $u$ solves a convex (or concave) fully nonlinear equation
 
$F(D^2 u, x) = 0$
 
which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.

Revision as of 21:06, 18 May 2011

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 \newcommand{\R}{\mathbb{R}}

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The surface quasi-geostrophic (SQG) equation consists of an evolution equation for a scalar function $\theta: \R^+ \times \R^2 \to \R$. In the inviscid case the equation is $$ \theta_t + u \cdot \nabla \theta = 0,$$ where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.

Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$

The equation is used as a toy model for the 3D Euler equation and Navier-Stokes. The main question is to determine whether the Cauchy problem is well posed. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions. The same comparison can be made between the supercritical SQG equation and Navier-Stokes.

For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.