Interior regularity results (local) and Semilinear equations: Difference between pages

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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).
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 $s<1/2$.


== Linear equations ==
== Some common semilinear equations ==


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 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]].


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]].
Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="OLC"/> <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>.


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.
=== 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
\[ u_t + (-\Delta)^s u = f(u). \]


* [[De Giorgi-Nash-Moser]]
The case $f(u) = u(1-u)$ corresponds to the KPP/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.
<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>
=== Burgers equation with fractional diffusion ===
is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).  
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"/>.


* [[Krylov-Safonov]]
=== [[Surface quasi-geostrophic equation]] ===
<math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>
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).


with <math>a_{ij}</math> unif elliptic, <math>b \in L^n</math> and <math>f \in L^n</math>, then the
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.
solution is <math>C^\alpha</math>


* [[Calderon-zygmund]]
=== Conservation laws with fractional diffusion ===
<math>a_{ij}(x) u_{ij} = f</math>
(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 $C^1$.


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>.
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"/>.


* [[Cordes-Nirenberg]]
=== Hamilton-Jacobi equation with fractional diffusion ===
<math>a_{ij}(x) u_{ij} = f</math>
It refers to the parabolic equation
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]


with <math>a_{ij}</math> close enough to the identity uniformly and $f \in L^\infty$,
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 $C^1$.
then $u$ is in $C^{1,\alpha}$


* [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)
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"/>.
<math>a_{ij}(x) u_{ij} = f</math>


with $a_{ij}$ close enough to the identity in a scale invariant Morrey
== References ==
norm in terms of $L^n$ and $f \in L^n$, then $u$ is in $C^{1,\alpha}$.
{{reflist|refs=
 
<ref name="OLC">{{Citation | last1=Ou | first1=Biao | last2=Li | first2=Congming | last3=Chen | first3=Wenxiong | title=Classification of solutions for an integral equation | url=http://dx.doi.org/10.1002/cpa.20116 | doi=10.1002/cpa.20116 | year=2006 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=59 | issue=3 | pages=330–343}}</ref>
($a_{ij} \in VMO$ is a particular case of this)
<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="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>
*[[Schauder]]
<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>
<math>a_{ij}(x) u_{ij} = f</math>
<ref name="K">{{Citation | last1=Kiselev | first1=A. | title=Nonlocal maximum principles for active scalars | year=to appear | journal=Advances in Mathematics}}</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>
with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$
<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>
 
<ref name="CS">{{Citation | last1=Cabre | first1=X. | last2=Sire | first2=Yannick | title=Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates | year=2010 | journal=Arxiv preprint arXiv:1012.0867}}</ref>
== Non linear equations ==
<ref name="CC"> {{Citation | last1=Cabré | first1=Xavier | last2=Cinti | first2=E. | title=Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian | year=2010 | journal=Discrete and Continuous Dynamical Systems (DCDS-A) | volume=28 | issue=3 | pages=1179–1206}} </ref>
 
<ref name="LF">{{Citation | last1=Frank | first1=R.L. | last2=Lenzmann | first2=E. | title=Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q+ Q-Q^{\alpha+1}= 0$ in $\R$ | year=2010 | journal=Arxiv preprint arXiv:1009.4042}}</ref>
* [[De Giorgi-Nash-Moser]]
<ref name="FQT">{{Citation | last1=Felmer | first1=P. | last2=Quaas | first2=A. | last3=Tan | first3=J. | title=Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian.}}</ref>
For any smooth strictly convex Lagrangian $L$, minimizers of functionals
<ref name="SV"> {{Citation | last1=Sire | first1=Yannick | last2=Valdinoci | first2=E. | title=Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result | publisher=[[Elsevier]] | year=2009 | journal=Journal of Functional Analysis | issn=0022-1236 | volume=256 | issue=6 | pages=1842–1864}} </ref>
 
<ref name="PSV">{{Citation | last1=Palatucci | first1=G. | last2=Valdinoci | first2=E. | last3=Savin | first3=O. | title=Local and global minimizers for a variational energy involving a fractional norm | year=2011 | journal=Arxiv preprint arXiv:1104.1725}}</ref>
$ \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]]
 
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]]
 
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]]
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 00:00, 7 February 2012

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, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained [1] [2] [3] [4] [5] [6] [7].

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 KPP/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 [8], 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 $s<1/2$ [9]. Still, if $s \in (0,1/2)$, the solution regularizes for large enough times[10][11].

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 $s > 1/2$ [12]. 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) [13] or the modulus of continuity approach [11].

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 [12]. The critical case $s=1/2$ was solved using the regularity results for drift-diffusion equations [13].

References

  1. ↑ Ou, Biao; Li, Congming; Chen, Wenxiong (2006), "Classification of solutions for an integral equation", Communications on Pure and Applied Mathematics 59 (3): 330–343, doi:10.1002/cpa.20116, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20116 
  2. ↑ Cabre, X.; Sire, Yannick (2010), "Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates", Arxiv preprint arXiv:1012.0867 
  3. ↑ Cabré, Xavier; Cinti, E. (2010), "Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian", Discrete and Continuous Dynamical Systems (DCDS-A) 28 (3): 1179–1206 
  4. ↑ Frank, R.L.; Lenzmann, E. (2010), "Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q+ Q-Q^{\alpha+1}= 0$ in $\R$", Arxiv preprint arXiv:1009.4042 
  5. ↑ Felmer, P.; Quaas, A.; Tan, J., Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian. 
  6. ↑ Sire, Yannick; Valdinoci, E. (2009), "Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result", Journal of Functional Analysis (Elsevier) 256 (6): 1842–1864, ISSN 0022-1236 
  7. ↑ Palatucci, G.; Valdinoci, E.; Savin, O. (2011), "Local and global minimizers for a variational energy involving a fractional norm", Arxiv preprint arXiv:1104.1725 
  8. ↑ 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 
  9. ↑ 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 
  10. ↑ 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 
  11. ↑ 11.0 11.1 Kiselev, A. (to appear), "Nonlocal maximum principles for active scalars", Advances in Mathematics 
  12. ↑ 12.0 12.1 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 
  13. ↑ 13.0 13.1 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 
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