Extension technique and Semilinear equations: Difference between pages

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The fractional Laplace operator $(-\Delta)^s$ on $\mathbb{R}^n$ is the Dirichlet-to-Neumann operator of a degenerate elliptic equation on the upper half-space $\mathbb{R}^{n+1}_+$.<ref name="CS"/> It is frequently used to localize problems involving the fractional Laplacian.  
An equation is called semilinear if it consists of the sum of a well understood linear term plus a lower order nonlinear term. For elliptic and parabolic equations, the two effective possibilities for the linear term is to be either the [[fractional Laplacian]] or the [[fractional heat equation]].


Let
Some equations which technically do not satisfy the definition above are still considered semilinear. For example evolution equations of the form
$$U:\mathbb{R}^n \times \mathbb{R}_+ \longrightarrow \mathbb{R}$$
\[ u_t + (-\Delta)^s u + H(x,u,Du) = 0 \]
be a function satisfying
can be thought of as semilinear equations even if $s<1/2$.
\begin{equation}
\label{eqn:Main}
\nabla \cdot (y^{1-2s} \nabla U(x,y)) = 0
\end{equation}
on the upper half-space, lying inside the appropriately weighted Sobolev space $\dot{H}(1-2s,\mathbb{R}^{n+1}_+)$. Then if we let $u(x) = U(x,0)$, we have
\begin{equation}
\label{eqn:Neumann}
(-\Delta)^s u(x) = -C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} \partial_y U(x,y).
\end{equation}
The energy associated with the operator in \eqref{eqn:Main} is
\begin{equation}
\label{eqn:Energy}
\int y^{1-2s} |\nabla U|^2 dx dy
\end{equation}


The weight $y^{1-2s}$, for $0<s<1$, lies inside the Muckenhoupt $A_2$ class of weights. It is known that degenerate 2nd order elliptic PDEs with these weights satisfy many of the usual properties of uniformly elliptic PDEs, such as the maximum principle, the [[De Giorgi-Nash-Moser]] regularity theory, the [[boundary Harnack inequality]], the Wiener criterion for regularity of a boundary point, all have suitable analogues.<ref name="FKS"/><ref name="FKJ1"/><ref name="FKJ2"/>
== Some common semilinear equations ==


The translation invariance of the operator in the $x$-directions can be applied to obtain higher regularity results and Liouville type properties.<ref name="CSS"/>
=== The most common elliptic equation in the world (provisional title) ===
Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs.
\[ (-\Delta)^s u = f(u). \]
If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].


==References==
Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, radial symmetry, etc...
 
=== Reaction diffusion equations ===
This general class refers to the equations we get by adding a zeroth order term to the right hand side of a heat equation. For the fractional case, it would look like
\[ u_t + (-\Delta)^s u = f(u). \]
 
The case $f(u) = u(1-u)$ corresponds to the Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior <ref name="CR"/>, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.
 
=== Burgers equation with fractional diffusion ===
It refers to the parabolic equation for a function on the real line $u:[0,+\infty) \times \R \to \R$,
\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
The equation is known to be well posed if $s \geq 1/2$ and to develop shocks if $s<1/2$ <ref name="KNS"/>. Still, if $s \in (0,1/2)$, the solution regularizes for large enough times<ref name="CCS"/><ref name="K"/>.
 
=== [[Surface quasi-geostrophic equation]] ===
It refers to the parabolic equation for a scalar function on the plane $\theta:[0,+\infty) \times \R^2 \to \R$,
\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]
where $u = R^\perp \theta$ (and $R$ is the Riesz transform).
 
The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. It is believed that solving the supercritical SQG equation could possibly help understand 3D Navier-Stokes equation.
 
=== Conservation laws with fractional diffusion ===
(aka "fractal conservation laws")
It refers to parabolic equations of the form
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
The Cauchy problem is known to be well posed classically if $s > 1/2$ <ref name="DI"/>. For $s<1/2$ there are viscosity solutions that are not $C^1$.
 
The critical case $s=1/2$ appears not to be written anywhere. However, it can be solved following the same method as for the Hamilton-Jacobi equations with fractional diffusion (below) <ref name="S"/> or the modulus of continuity approach <ref name="K"/>.
 
=== Hamilton-Jacobi equation with fractional diffusion ===
It refers to the parabolic equation
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
 
The Cauchy problem is known to be well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.
 
The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]] <ref name="DI"/>. The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]] <ref name="S"/>.
 
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=An extension problem related to the fractional Laplacian | url=http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 | doi=10.1080/03605300600987306 | year=2007 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=32 | issue=7 | pages=1245–1260}}</ref>
<ref name="KNS">{{Citation | last1=Kiselev | first1=Alexander | last2=Nazarov | first2=Fedor | last3=Shterenberg | first3=Roman | title=Blow up and regularity for fractal Burgers equation | year=2008 | journal=Dynamics of Partial Differential Equations | issn=1548-159X | volume=5 | issue=3 | pages=211–240}}</ref>
<ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion | url=http://dx.doi.org/10.1016/j.aim.2010.09.007 | doi=10.1016/j.aim.2010.09.007 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=2020–2039}}</ref>
<ref name="FKS">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Serapioni | first3=Raul P. | title=The local regularity of solutions of degenerate elliptic equations | url=http://dx.doi.org/10.1080/03605308208820218 | doi=10.1080/03605308208820218 | year=1982 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=7 | issue=1 | pages=77–116}}</ref>
<ref name="CCS">{{Citation | last1=Chan | first1=Chi Hin | last2=Czubak | first2=Magdalena | last3=Silvestre | first3=Luis | title=Eventual regularization of the slightly supercritical fractional Burgers equation | url=http://dx.doi.org/10.3934/dcds.2010.27.847 | doi=10.3934/dcds.2010.27.847 | year=2010 | journal=Discrete and Continuous Dynamical Systems. Series A | issn=1078-0947 | volume=27 | issue=2 | pages=847–861}}</ref>
<ref name="FKJ1">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Jerison | first3=David | title=Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) | publisher=Wadsworth | series=Wadsworth Math. Ser. | year=1983 | chapter=Boundary behavior of solutions to degenerate elliptic equations | pages=577–589}}</ref>
<ref name="K">{{Citation | last1=Kiselev | first1=A. | title=Regularity and blow up for active scalars | url=http://dx.doi.org/10.1051/mmnp/20105410 | doi=10.1051/mmnp/20105410 | year=2010 | journal=Mathematical Modelling of Natural Phenomena | issn=0973-5348 | volume=5 | issue=4 | pages=225–255}}</ref>
<ref name="FKJ2">{{Citation | last1=Fabes | first1=Eugene B. | last2=Jerison | first2=David | last3=Kenig | first3=Carlos E. | title=The Wiener test for degenerate elliptic equations | url=http://www.numdam.org/item?id=AIF_1982__32_3_151_0 | year=1982 | journal=[[Annales de l'Institut Fourier|Université de Grenoble. Annales de l'Institut Fourier]] | issn=0373-0956 | volume=32 | issue=3 | pages=151–182}}</ref>
<ref name="DI">{{Citation | last1=Droniou | first1=Jérôme | last2=Imbert | first2=Cyril | title=Fractal first-order partial differential equations | url=http://dx.doi.org/10.1007/s00205-006-0429-2 | doi=10.1007/s00205-006-0429-2 | year=2006 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | volume=182 | issue=2 | pages=299–331}}</ref>
<ref name="CR">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire | url=http://dx.doi.org/10.1016/j.crma.2009.10.012 | doi=10.1016/j.crma.2009.10.012 | year=2009 | journal=Comptes Rendus Mathématique. Académie des Sciences. Paris | issn=1631-073X | volume=347 | issue=23 | pages=1361–1366}}</ref>
}}
}}

Revision as of 18:14, 31 May 2011

An equation is called semilinear if it consists of the sum of a well understood linear term plus a lower order nonlinear term. For elliptic and parabolic equations, the two effective possibilities for the linear term is to be either the fractional Laplacian or the fractional heat equation.

Some equations which technically do not satisfy the definition above are still considered semilinear. For example evolution equations of the form \[ u_t + (-\Delta)^s u + H(x,u,Du) = 0 \] can be thought of as semilinear equations even if $s<1/2$.

Some common semilinear equations

The most common elliptic equation in the world (provisional title)

Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs. \[ (-\Delta)^s u = f(u). \] If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^p$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard bootstrapping.

Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, radial symmetry, etc...

Reaction diffusion equations

This general class refers to the equations we get by adding a zeroth order term to the right hand side of a heat equation. For the fractional case, it would look like \[ u_t + (-\Delta)^s u = f(u). \]

The case $f(u) = u(1-u)$ corresponds to the Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior [1], etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.

Burgers equation with fractional diffusion

It refers to the parabolic equation for a function on the real line $u:[0,+\infty) \times \R \to \R$, \[ u_t + u \ u_x + (-\Delta)^s u = 0 \] The equation is known to be well posed if $s \geq 1/2$ and to develop shocks if $s<1/2$ [2]. Still, if $s \in (0,1/2)$, the solution regularizes for large enough times[3][4].

Surface quasi-geostrophic equation

It refers to the parabolic equation for a scalar function on the plane $\theta:[0,+\infty) \times \R^2 \to \R$, \[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \] where $u = R^\perp \theta$ (and $R$ is the Riesz transform).

The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. It is believed that solving the supercritical SQG equation could possibly help understand 3D Navier-Stokes equation.

Conservation laws with fractional diffusion

(aka "fractal conservation laws") It refers to parabolic equations of the form \[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\] The Cauchy problem is known to be well posed classically if $s > 1/2$ [5]. For $s<1/2$ there are viscosity solutions that are not $C^1$.

The critical case $s=1/2$ appears not to be written anywhere. However, it can be solved following the same method as for the Hamilton-Jacobi equations with fractional diffusion (below) [6] or the modulus of continuity approach [4].

Hamilton-Jacobi equation with fractional diffusion

It refers to the parabolic equation \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]

The Cauchy problem is known to be well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.

The subcritical case $s>1/2$ can be solved with classical bootstrapping [5]. The critical case $s=1/2$ was solved using the regularity results for drift-diffusion equations [6].

References

  1. Cabré, Xavier; Roquejoffre, Jean-Michel (2009), "Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire", Comptes Rendus Mathématique. Académie des Sciences. Paris 347 (23): 1361–1366, doi:10.1016/j.crma.2009.10.012, ISSN 1631-073X, http://dx.doi.org/10.1016/j.crma.2009.10.012 
  2. Kiselev, Alexander; Nazarov, Fedor; Shterenberg, Roman (2008), "Blow up and regularity for fractal Burgers equation", Dynamics of Partial Differential Equations 5 (3): 211–240, ISSN 1548-159X 
  3. Chan, Chi Hin; Czubak, Magdalena; Silvestre, Luis (2010), "Eventual regularization of the slightly supercritical fractional Burgers equation", Discrete and Continuous Dynamical Systems. Series A 27 (2): 847–861, doi:10.3934/dcds.2010.27.847, ISSN 1078-0947, http://dx.doi.org/10.3934/dcds.2010.27.847 
  4. 4.0 4.1 Kiselev, A. (2010), "Regularity and blow up for active scalars", Mathematical Modelling of Natural Phenomena 5 (4): 225–255, doi:10.1051/mmnp/20105410, ISSN 0973-5348, http://dx.doi.org/10.1051/mmnp/20105410 
  5. 5.0 5.1 Droniou, Jérôme; Imbert, Cyril (2006), "Fractal first-order partial differential equations", Archive for Rational Mechanics and Analysis 182 (2): 299–331, doi:10.1007/s00205-006-0429-2, ISSN 0003-9527, http://dx.doi.org/10.1007/s00205-006-0429-2 
  6. 6.0 6.1 Silvestre, Luis (2011), "On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion", Advances in Mathematics 226 (2): 2020–2039, doi:10.1016/j.aim.2010.09.007, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2010.09.007