Fractional Laplacian and Active scalar equation: Difference between pages

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The fractional Laplacian $(-\Delta)^s$ is a classical operator which gives the standard Laplacian when $s=1$. One can think of $-(-\Delta)^s$ as the most basic  [[elliptic linear integro-differential operator]] of order $2s$ and can be defined in several equivalent ways (listed below). A range of powers of particular interest is $s \in (0,1)$, in which case for $u \in \mathcal{S}(\mathbb{R}^n)$ we can write the operator as
A general class of equations is often referred to as ''active scalars''. It consists of solving the Cauchy problem for the transport equation
\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.


\[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\]
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.


where $c_{n,s}$ is a universal constant and $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$. This particular expression shows that in this range of $s$ the operator enjoys the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)^s u(x) \geq 0$, with equality only if $u$ is constant. From this monotonicity, a [[comparison principle]] can be derived for equations involving the fractional Laplacian.
For their applications to 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]].


== Definitions ==
== General properties ==
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...


All the definitions below are equivalent.
On the other hand, the following $s$-dependent energy is preserved by the flow:
\[||\theta||_{\dot H^{-s}}^2 = \int \theta (-\Delta)^{-s} \theta \ dx.\]


=== As a Fourier multiplier ===
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 fractional Laplacian is the operator with symbol $|\xi|^{2s}$. In other words, the following formula holds
\[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\]
for any function (or tempered distribution) for which the right hand side makes sense.


This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.
== 2D Euler (well posedness) ==
The usual Euler equation refers to the system
\begin{align}
\partial_t u + u \cdot \nabla u &= -\nabla p \\
\mathrm{div} \ u &= 0
\end{align}
where $u$ is a vector valued function and $p$ is a scalar function.


=== Via the heat semigroup ===
In 2D, the vorticity $\omega(x,y) = \partial_x u_2 - \partial_y u_1$ satisfies the active scalar equation
Recall the numerical formula
\[ \partial_t \omega + u \cdot \nabla \omega = 0 \]
$$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\infy\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$
where $u = \nabla^\perp (-\Delta)^{-1} \omega$.
valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=|\xi|^2$ and using the Fourier transform definition of the fractional Laplacian we get
$$(-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}.$$
Here $v(x,t)=e^{t\Delta}f(x)$ is the solution to the heat equation on $\mathbb{R}^n$ with initial temperature $f$, namely,
$$\begin{cases}
\partial_tv=\Delta v,&\hbox{for}~(x,t)\in\mathbb{R}^n\times(0,\infty)\\
v(x,0)=f(x),&\hbox{for}~x\in\mathbb{R}^n.
\end{cases}$$
See <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>.


=== From functional calculus ===
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.
Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.


This definition is not as useful for practical applications, since it does not provide any explicit formula.
== Inviscid [[surface quasi-geostrophic equation]] ==
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 $BMO$.


=== As a singular integral ===
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.
If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula
\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\]
 
Where $c_{n,s}$ is a constant depending on dimension and $s$.
 
This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.
 
=== As a generator of a [[Levy process]] ===
The operator can be defined as the generator of $\alpha$-stable Lévy processes. More precisely, if $X_t$ is the isotropic $\alpha$-stable Lévy process starting at zero and $f$ is a smooth function, then
\[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]
 
This definition is important for applications to probability.
 
== Inverse operator ==
The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the ''Riesz potential'':
\[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\]
which holds as long as $u$ is integrable enough for the right hand side to make sense.
 
== Heat kernel ==
The fractional heat kernel $p(t,x)$ is the fundamental solution to the [[fractional heat equation]]. It is the function which solves the equation
\begin{align*}
p(0,x) &= \delta_0 \\
p_t(t,x) + (-\Delta)^s p &= 0
\end{align*}
 
The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables, but the following inequalities are known to hold for some constant $C$
\[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]
 
Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
\[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]
 
== Poisson kernel ==
Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem
\begin{align*}
u(x) &= g(x) \qquad \text{if } x \notin B_1 \\
(-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1.
\end{align*}
 
The solution can be computed explicitly using the Poisson kernel
\[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
where<ref name="R"/>
\[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]
 
The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')<ref name="L"/>.
 
== Green's function for the ball ==
For a function $g \in L^2(B_1)$, there exists a unique function $u \in H^s(\R^n)$ such that
\begin{align*}
u(x) &= 0 && \text{if } x \notin B_1 \\
(-\Delta)^s u &= g(x) && \text{if } x \in B_1.
\end{align*}
 
The solution is given explicitly using the Green's function,
\[ u(x) = \int_{B_1} G_{B_1}(x, y) g(y) \mathrm d y, \]
where<ref name="R"/>
\[ G_{B_1}(x, y) = C_{n,s} |x - y|^{2 s - n} \int_0^{r_0(x, y)} \frac{r^{s-1}}{(r+1)^{n/2}} \, \mathrm d r \]
with
\[ r_0(x, y) = \frac{(1 - |x|^2) (1 - |y|^2)}{|x - y|^2} . \]
The above formula holds for all $s \in (0, 1)$ also for $n = 1$.<ref name="BGR"/>
 
== Regularity issues ==
Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This follows from the smoothness of the Poisson kernel for balls.
 
More generally, one has the estimate
\[\|u\|_{C^{\alpha+2s}(B_{1/2})}\leq C\left(
\|(-\Delta)^s u\|_{C^{\alpha}(B_1)}+\|u\|_{L^{\infty}(B_1)}+\int_{\R^n\setminus B_1}|u(y)|\frac{dy}{|y|^{n+2s}}\right)\]
for any $\alpha\geq0$ such that $\alpha+2s$ is not an integer.
 
=== Full space regularization of the Riesz potential ===
If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions ''up to $2s$ derivatives''. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of <ref name="S"/>, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$, $s>0$. However, if $f$ belongs to $L^p$ then it does not follow that $u\in W^{2s,p}$; this is true only for $p\geq2$. For $1<p<2$ one only have $u\in B^{2s}_{p,2}\supset W^{2s,p}$ ---see Chapter V in Stein<ref name="Stein"/>.
 
=== Boundary regularity ===
From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel).
 
Even if $u$ is not $C^\infty$ up to the boundary, we have the following: consider the solution $u$ to the Dirichlet problem  
\[\left\{ \begin{array}{rcll}
(-\Delta)^s u &=&g&\textrm{in }\Omega \\
u&=&0&\textrm{in }\R^n\backslash \Omega.
\end{array}\right.\]
If $\Omega$ is $C^\infty$, then
\[g\in C^\infty(\overline\Omega)\qquad \Longrightarrow \qquad u/d^s\in C^\infty(\overline\Omega),\]
where $d(x)$ is (a smoothed version of) the distance to $\partial\Omega$; see <ref name="Grubb"/> and also <ref name="RS"/>.
 
If $\Omega$ is $C^{2,\alpha}$ and $g$ is $C^\alpha$, then $u/d^s$ is $C^{\alpha+s}$ up to the boundary <ref name="RS-K"/>.
 
Related to this, if $g$ is not bounded but only in $L^p(\Omega)$ then $u\in L^q$ with $q=\frac{np}{n-2ps}$ in case $p<n/(2s)$, while $u\in L^\infty(\Omega)$ in case $p>n/(2s)$ ---see for example Proposition 1.4 in <ref name="RS2"/>.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="BGR">{{Citation | last1=Blumenthal | first1=R. M. | last2=Getoor | first2=R. K. | last3=Ray | first3=D. B. | title=On the distribution of first hits for the symmetric stable processes | url=http://www.jstor.org/stable/1993561 | year=1961 | journal=Trans. Amer. Math. Soc. | issn=0002-9947 | volume=99 | pages=540–554}}</ref>
<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>
<ref name="L">{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1972}}</ref>
<ref name="R">{{Citation | last1=Riesz | first1=M. | title=Intégrales de Riemann-Liouville et potentiels | url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article | year=1938 | journal=Acta Sci. Math. Szeged | issn=0001-6969 | volume=9 | issue=1 | pages=1–42}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
<ref name="RS">{{Citation  | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The Dirichlet problem for the fractional Laplacian: regularity up to the boundary | url=http://arxiv.org/abs/1207.5985 | year=2012 | journal=[[J. Math. Pures Appl.]] | volume=101 | pages=275-302 }}</ref>
<ref name="Stein">{{Citation | last1=Stein | first1=E. | title=Singular Integrals And Differentiability Properties Of Functions | publisher=[[Princeton Mathematical Series]] | year=1970}}</ref>
<ref name="RS2">{{Citation  | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The extremal solution for the fractional Laplacian | url=http://arxiv.org/abs/1305.2489 | year=2013 | journal=[[Calc. Var. Partial Differential Equations]] | pages=to appear }}</ref>
<ref name="Grubb">{{Citation  | last1=Grubb | first1=G. | title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators | url=http://arxiv.org/abs/1310.0951 | year=2014 | journal=[[arXiv]] | pages=1-43 }}</ref>
<ref name="RS-K">{{Citation  | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=Boundary regularity for fully nonlinear integro-differential equations | year=2014 | journal=[[preprint arXiv]] }}</ref>
}}
}}
<ref name="Stinga">{{Citation  | last1=Stinga | first1=P. R.| title=Fractional powers of second order partial differential operators: extension problem and regularity theory | url=https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf | year=2010 | journal=[[PhD. Thesis - Universidad Aut\'onoma de Madrid]]}}</ref>

Revision as of 18:46, 13 March 2012

A general class of equations is often referred to as active scalars. It consists of solving the Cauchy problem for the transport equation \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.

For their applications to 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.

General properties

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: \[||\theta||_{\dot H^{-s}}^2 = \int \theta (-\Delta)^{-s} \theta \ dx.\]

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 $s \in [0,1]$.[1]

2D Euler (well posedness)

The usual Euler equation refers to the system \begin{align} \partial_t u + u \cdot \nabla u &= -\nabla p \\ \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 \[ \partial_t \omega + u \cdot \nabla \omega = 0 \] where $u = \nabla^\perp (-\Delta)^{-1} \omega$.

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.

Inviscid surface quasi-geostrophic equation

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 $BMO$.

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.

References

  1. Chae, D.; Gancedo, F.; Córdoba, D.; Constantin, Peter; Wu, Jun (2011), "Generalized surface quasi-geostrophic equations with singular velocities", Arxiv preprint arXiv:1101.3537