Linear integro-differential operator and Semilinear equations: Difference between pages

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The linear integro-differential operators that we consider ''in this wiki'' are the generators of [[Levy processes]]. According to the Levy-Kintchine formula, they have the general form
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]].


\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
Some equations which technically do not satisfy the definition above are still considered semilinear. For example evolution equations of the form
where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
\[ u_t + (-\Delta)^s u + H(x,u,Du) = 0 \]
\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
can be thought of as semilinear equations even if $s<1/2$.


The above definition is very general. Many theorems, and in particular regularity theorems, require extra assumptions in the kernels $K$. These assumptions restrict the study to certain sub-classes of linear operators. The simplest of all is the [[fractional Laplacian]]. We list below several extra assumptions that are usually made.
== Some common semilinear equations ==


== Absolutely continuous measure ==
=== 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 has 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]].


In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
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...


We keep this assumption in all the examples below.
=== 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
\[ u_t + (-\Delta)^s u = f(u). \]


== Purely integro-differential operator ==
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, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.


In this case we neglect the local part of the operator
=== Burgers equation with fractional diffusion ===
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]
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"/>.


== Symmetric kernels ==
=== [[Surface quasi-geostrophic equation]] ===
If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.
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).


In the purely integro-differentiable case, it reads as
The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem.
\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]


The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
=== 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 $C^1$.


== Translation invariant operators ==
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"/>.
In this case, all coefficients are independent of $x$.
\[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]


== The fractional Laplacian ==
=== Hamilton-Jacobi equation with fractional diffusion ===
It refers to the parabolic equation
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]


The [[fractional Laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.
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 $C^1$.
 
\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]
 
== Stable operators ==
 
These are the operators whose kernel is homogeneous in $y$
\[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\]
They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure.
 
== Uniformly elliptic of order $s$ ==
 
This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order.
\[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]
 
The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
 
An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.
 
== Smoothness class $k$ of order $s$ ==
This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded
\[ |\partial_y^r K(x,y)| \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]
 
== Order strictly below one ==
 
If a non symmetric kernel $K$ satisfies the extra local integrability assumption
\[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
then the extra gradient term is not necessary in order to define the operator.
 
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]
 
The modification in the integro-differential part of the operator becomes an extra drift term.
 
A uniformly elliptic operator of order $s<1$ satisfies this condition.
 
== Order strictly above one ==
 
If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
\[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
then the gradient term in the integral can be taken global instead of being cut off in the unit ball.
 
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]
 
The modification in the integro-differential part of the operator becomes an extra drift term.
 
A uniformly elliptic operator of order $s>1$ satisfies this condition.
 
== More singular/irregular kernels ==
 
The concept of uniform ellipticity can be relaxed in various ways. The following family of kernels was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />
 
For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \R^d \to \R$ is assumed to satisfy the following assumptions.
* $K(h) \geq 0$ for all $h\in\R^d$.
This is a basic assumption for the integral operator to be a legit diffusion operator.
 
* For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]
This assumption is more general than $K(y) \leq (2-\alpha) \Lambda |y|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.
 
* For every $r>0$, there exists a set $A_r$ such that
** $A_r \subset B_{2r} \setminus B_r$.
** $A_r$ is symmetric in the sense that $A_r = -A_r$.
** $|A_r| \geq \mu |B_{2r} \setminus B_r|$.
** $K(h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.
Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_r|.
\]
 
This assumption says that the lower bound $K(y) \geq (2-\alpha) \lambda |y|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.
 
* For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}. \]
 
This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$.
 
== Indexed by a matrix ==
In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
\[ K_A(y) =  \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$.
 
== Second order elliptic operators as limits of purely integro-differential ones ==
 
Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators
 
\[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \]
 
define in the limit $\sigma \to 2^-$ a second order linear elliptic operator (possibly degenerate). This can be checked for any fixed $C^2$ function $u$ by a straightforward computation using the second order Taylor expansion. A class of kernels that is big enough to recover all translation invariant elliptic operators of the form  $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels
 
\[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\]
 
where $A$ is an invertible symmetric matrix.
 
== Characterization via global maximum principle ==
 
A bounded linear operator
 
\[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \]
 
is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have
 
\[ (Lu)(x_0) \leq 0 \]
 
It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège <ref name="C65"/> <ref name="C64"/>: if $L$ satisfies the global maximum principle then it has the form
 
\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
 
where again $A(x)$ is a nonnegative matrix for all $x$, $c(x)\leq 0$ and $\mu_x$ is a nonnegative measure for all $x$ satisfying
 
\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
 
and $A(x),c(x)$ and $b(x)$ are bounded.
 
 
== See also ==
 
* [[Fractional Laplacian]]
* [[Levy processes]]
* [[Dirichlet form]]


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 ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="C64">{{Citation | last1=Courrège | first1=Philippe | title=Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine | year=1964 | journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=88 | pages=3–30}}</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="C65">{{Citation | last1=Courrège | first1=P. | title=Sur la forme intégro-différentielle des opéateurs de  $C_k^\infty(\mathbb{R}^n)$  dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum | journal=Sém. Théorie du potentiel (1965/66) Exposé | volume=2}}</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="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</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="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="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>
}}
}}

Revision as of 18:05, 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 has 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, 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 $s<1/2$ [1]. Still, if $s \in (0,1/2)$, the solution regularizes for large enough times[2][3].

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.

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 $s > 1/2$ [4]. 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) [5] or the modulus of continuity approach [3].

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 [4]. The critical case $s=1/2$ was solved using the regularity results for drift-diffusion equations [5].

References

  1. 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 
  2. 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 
  3. 3.0 3.1 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 
  4. 4.0 4.1 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 
  5. 5.0 5.1 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