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| 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]].
| | This is a list of nonlocal equations that appear in this wiki. |
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| Some equations which technically do not satisfy the definition above are still considered semilinear. For example evolution equations of the form
| | == Linear equations == |
| \[ u_t + (-\Delta)^s u + H(x,u,Du) = 0 \] | | === Stationary linear equations from Levy processes === |
| can be thought of as semilinear equations even if $s<1/2$.
| | \[ Lu = 0 \] |
| | where $L$ is a [[linear integro-differential operator]]. |
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| == Some common semilinear equations == | | === parabolic linear equations from Levy processes === |
| | \[ u_t = Lu \] |
| | where $L$ is a [[linear integro-differential operator]]. |
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| | === [[Drift-diffusion equations]] === |
| | \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\] |
| | where $b$ is a given vector field. |
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| | == [[Semilinear equations]] == |
| === Stationary equations with zeroth order nonlinearity === | | === Stationary equations with zeroth order nonlinearity === |
| 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.
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| \[ (-\Delta)^s u = f(u). \] | | \[ (-\Delta)^s u = f(u). \] |
| If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^\infty$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]].
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| Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, boundedness, 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"/>.
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| === Reaction diffusion equations === | | === 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
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| \[ u_t + (-\Delta)^s u = f(u). \] | | \[ u_t + (-\Delta)^s u = f(u). \] |
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| 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.
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| === Burgers equation with fractional diffusion === | | === Burgers equation with fractional diffusion === |
| It refers to the parabolic equation for a function on the real line $u:[0,+\infty) \times \R \to \R$,
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| \[ u_t + u \ u_x + (-\Delta)^s u = 0 \] | | \[ u_t + u \ u_x + (-\Delta)^s u = 0 \] |
| 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"/>.
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| === [[Surface quasi-geostrophic equation]] === | | === [[Surface quasi-geostrophic equation]] === |
| It refers to the parabolic equation for a scalar function on the plane $\theta:[0,+\infty) \times \R^2 \to \R$,
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| \[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \] | | \[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \] |
| where $u = R^\perp \theta$ (and $R$ is the Riesz transform).
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| 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.
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| === Conservation laws with fractional diffusion === | | === Conservation laws with fractional diffusion === |
| (aka "fractal conservation laws")
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| It refers to parabolic equations of the form
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| \[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\] | | \[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\] |
| 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 $C^1$.
| | === Hamilton-Jacobi equation with fractional diffusion === |
| | \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\] |
| | === [[Keller-Segel equation]] === |
| | \[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\] |
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| | == Quasilinear or [[fully nonlinear integro-differential equations]] == |
| | === [[Bellman equation]] === |
| | \[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \] |
| | where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$. |
| | === [[Isaacs equation]] === |
| | \[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] |
| | where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$. |
| | === [[obstacle problem]] === |
| | For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies |
| | \begin{align} |
| | u &\geq \varphi \qquad \text{everywhere in the domain } D,\\ |
| | Lu &\leq 0 \qquad \text{everywhere in the domain } D,\\ |
| | Lu &= 0 \qquad \text{wherever } u > \varphi. |
| | \end{align} |
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| 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"/>. | | === [[Nonlocal minimal surfaces ]] === |
| | The set $E$ satisfies. |
| | \[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\] |
| | === [[Nonlocal porous medium equation]] === |
| | \[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\] |
| | Or |
| | \[ u_t +(-\Delta)^{s}(u^m) = 0. \] |
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| === Hamilton-Jacobi equation with fractional diffusion === | | == Inviscid equations == |
| It refers to the parabolic equation
| | === [[Surface quasi-geostrophic equation|Inviscid SQG]]=== |
| \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\] | | \[ \theta_t + u \cdot \nabla \theta = 0,\] |
| | where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$. |
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| For Lipschitz initial data, the Cauchy problem always has a viscosity solution which is Lipschitz in space.<ref name="DI"/> The problem is well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.
| | === [[Active scalar equation]] (from fluid mechanics) === |
| | \[ \theta_t + u \cdot \nabla \theta = 0,\] |
| | where $u = \nabla^\perp K \ast \theta$. |
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| 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"/>
| | === [[Aggregation equation]] === |
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| == References ==
| | \[ \theta_t + \mathrm{div}(\theta \ u) = 0,\] |
| {{reflist|refs=
| | where $u = \nabla K \ast \theta$. |
| <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>
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| <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>
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| <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>
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| <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>
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| <ref name="K">{{Citation | last1=Kiselev | first1=A. | title=Nonlocal maximum principles for active scalars | year=to appear | journal=Advances in Mathematics}}</ref>
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| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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This is a list of nonlocal equations that appear in this wiki.
Linear equations
Stationary linear equations from Levy processes
\[ Lu = 0 \]
where $L$ is a linear integro-differential operator.
parabolic linear equations from Levy processes
\[ u_t = Lu \]
where $L$ is a linear integro-differential operator.
\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\]
where $b$ is a given vector field.
Stationary equations with zeroth order nonlinearity
\[ (-\Delta)^s u = f(u). \]
Reaction diffusion equations
\[ u_t + (-\Delta)^s u = f(u). \]
Burgers equation with fractional diffusion
\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]
Conservation laws with fractional diffusion
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
Hamilton-Jacobi equation with fractional diffusion
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]
\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies
\begin{align}
u &\geq \varphi \qquad \text{everywhere in the domain } D,\\
Lu &\leq 0 \qquad \text{everywhere in the domain } D,\\
Lu &= 0 \qquad \text{wherever } u > \varphi.
\end{align}
The set $E$ satisfies.
\[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]
\[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\]
Or
\[ u_t +(-\Delta)^{s}(u^m) = 0. \]
Inviscid equations
\[ \theta_t + u \cdot \nabla \theta = 0,\]
where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.
\[ \theta_t + u \cdot \nabla \theta = 0,\]
where $u = \nabla^\perp K \ast \theta$.
\[ \theta_t + \mathrm{div}(\theta \ u) = 0,\]
where $u = \nabla K \ast \theta$.