Active scalar equation and De Giorgi-Nash-Moser theorem: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Luis
No edit summary
 
imported>Luis
(Created page with "The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. T...")
 
Line 1: Line 1:
A general class of equations is often referred to as ''active scalars''. It consists of solving the Cauchy problem for the transport equation
The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form.
\begin{align}
\theta(x,t) &= \theta_0(x) \\
\partial_t \theta + u \cdot \nabla \theta &= 0
\end{align}
where the vector field $u$ is related to $\theta$ by some operator.  


The case $u = \nabla^\perp (-\Delta)^{-1} \theta$, in two space dimensions, corresponds to the vorticity formulation of the 2D Euler equation. The case $u = \nabla^\perp (-\Delta)^{-1/2} \theta$, in two space dimensions, corresponds to the inviscid [[surface quasi-geostrophic equation]]. If we consider the full range of exponents $u = \nabla^\perp (-\Delta)^{-s} \theta$, the equation is known to be well posed in the classical sense if $s \geq 1$. For any $s<1$, the possible break down of classical solutions in finite time is an open problem.
The equation is
\[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \]
in the elliptic case, or
\[ u_t = \mathrm{div} A(x,t) \nabla u(x). \]
Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$,
\[ \langle A v,v \rangle \geq \lambda |v|^2,\]
for every $v \in \R^n$, uniformly in space and time.


As a model for the equations of fluid dynamics, the vector field $u$ is taken to be divergence free. The opposite case when $u$ is the gradient of a potential (or strictly speaking its dual equation) is studied in the context of the [[aggregation equation]] and the [[nonlocal porous medium equation]].
The corresponding result in non divergence form is [[Krylov-Safonov theorem]].


== General properties ==
For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].
In two space dimensions, under the general choice $u = \nabla^\perp (-\Delta)^{-s} \theta$, the vector field $u$ is divergence free. Therefore the transport equation enjoys all properties of divergence free flows: all $L^p$ norms of $\theta$ are conserved, the distribution function of $\theta$ is conserved, the set of points where the trajectories cross has measure zero, etc...


On the other hand, the following $s$-dependent energy is preserved by the flow:
== Elliptic version ==
\[||\theta||_{\dot H^{-s}}^2 = \int \theta (-\Delta)^{-s} \theta \ dx.\]
For the result in the elliptic case, we assume that the equation
\[ \mathrm{div} A(x) \nabla u(x) = 0 \]
is satisfied in the unit ball $B_1$ of $\R^n$.
===Holder estimate===
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and
\[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.


The equation reduces to 2D Euler if $s=1$, and to inviscid SQG if $s=1/2$. The operator determining the velocity is more singular the smaller $s$ is. On the other hand, the conserved energy becomes a stronger quantity. The problem is known to be locally well posed for $s \in [0,1]$.<ref name="CGCCW"/>
The result can be scaled to balls of arbitrary radius $r>0$ to obtain
\[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]


There exists no differential operator which maps a scalar function $\theta$ to a '''divergence free''' vector field $u$, which commutes with translations and rotations, for which the equation is known to develop singularities in finite time. This is an open problem even for high order operators like $u = \nabla^\perp \Delta^{10} \theta$.
Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$.


== 2D Euler (well posedness) ==
===Harnack inequality===
The usual Euler equation refers to the system
The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$:
\begin{align}
\[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\]
\partial_t u + u \cdot \nabla u &= -\nabla p \\
The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.
\mathrm{div} \ u &= 0
\end{align}
where $u$ is a vector valued function and $p$ is a scalar function.


In 2D, the vorticity $\omega(x,y) = \partial_x u_2 - \partial_y u_1$ satisfies the active scalar equation
===Minimizers of convex functionals===
\[ \partial_t \omega + u \cdot \nabla \omega = 0 \]
The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals
where $u = \nabla^\perp (-\Delta)^{-1} \omega$.
\[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\]
are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by [[bootstrapping]] with the [[Schauder estimates]] and the smoothness of $F$.


The equation is (borderline) well posed for the following reason. The $L^\infty$ norm of $\omega$ is clearly preserved since it is a transport equation. In order to obtain higher regularity estimates on $\omega$ we need to estimate the rate by which the trajectories of the flow by $u$ approach each other. The most usual way to do this is by estimating the Lipschitz norm of $u$. The fact that $\omega \in L^\infty$ uniformly in time does not immediately imply that $u$ is Lipschitz. Instead it implies the borderline weaker condition $u \in LogLip$. Thus, in particular $u$ satisfies the Osgood condition and the flow trajectories are uniquely defined. From this property of the flow one can easily derive higher regularity estimates for $\omega$ that grow doubly exponentially in time.
Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made.


== Inviscid [[surface quasi-geostrophic equation]] ==
== Parabolic version ==
The inviscid SQG equation corresponds to the choice $u = \nabla^{\perp} (-\Delta)^{-1/2} \theta$. In this case the velocity is given by an operator of order zero applied to $\theta$, which always gives a divergence free drift. From the $L^\infty$ a priori estimate on $\theta$, the vector field $u$ stays bounded in $BMO$.
For the result in the parabolic case, we assume that the equation
\[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \]
is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.
===Holder estimate===
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and
\[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.


This is a case which attracts a lot of interest. The known results coincide with the general case of active scalar equations for $s$ in the range $(0,1)$. The classical well posedness of the equation for large time is still an open problem.
===Harnack inequality===
The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time:
\[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]


== References ==
===Gradient flows===
{{reflist|refs=
The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth.
<ref name="CGCCW">{{Citation | last1=Chae | first1=D. | last2=Gancedo | first2=F. | last3=Córdoba | first3=D. | last4=Constantin | first4=Peter | last5=Wu | first5=Jun | title=Generalized surface quasi-geostrophic equations with singular velocities | year=2011 | journal=Arxiv preprint arXiv:1101.3537}}</ref>
\[ u_t + \partial_u J[u] = u_t + \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\]
}}
The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.

Revision as of 14:49, 14 March 2012

The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form.

The equation is \[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \] in the elliptic case, or \[ u_t = \mathrm{div} A(x,t) \nabla u(x). \] Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$, \[ \langle A v,v \rangle \geq \lambda |v|^2,\] for every $v \in \R^n$, uniformly in space and time.

The corresponding result in non divergence form is Krylov-Safonov theorem.

For nonlocal equations, there are analogous results both for Holder estimates and the Harnack inequality.

Elliptic version

For the result in the elliptic case, we assume that the equation \[ \mathrm{div} A(x) \nabla u(x) = 0 \] is satisfied in the unit ball $B_1$ of $\R^n$.

Holder estimate

The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and \[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\] The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.

The result can be scaled to balls of arbitrary radius $r>0$ to obtain \[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]

Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$.

Harnack inequality

The Harnack inequality says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$: \[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\] The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.

Minimizers of convex functionals

The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals \[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\] are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by bootstrapping with the Schauder estimates and the smoothness of $F$.

Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made.

Parabolic version

For the result in the parabolic case, we assume that the equation \[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \] is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.

Holder estimate

The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and \[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\] The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.

Harnack inequality

The Harnack inequality says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time: \[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]

Gradient flows

The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth. \[ u_t + \partial_u J[u] = u_t + \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\] The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.

Retrieved from "https://web.ma.utexas.edu/mediawiki/index.php?title=Active_scalar_equation&oldid=1129"