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\begin{center}
\textbf{Examples}
\end{center}

\bigskip Definition: Linear \ $au_{1}+bu_{2}$ is also a solution

\begin{itemize}
\item semi-linear, quasi-linear

\item $\frac{\partial u}{\partial x}+a\frac{\partial u}{\partial y}=0\qquad 
\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0$

\item $\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=0\qquad 
\frac{\partial u}{\partial t}+x\frac{\partial u}{\partial x}=0$

\item $\frac{\partial u}{\partial x}+u\frac{\partial u}{\partial y}=0\qquad 
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$

\item $\vartriangle \!\!u=\frac{\partial ^{2}u}{\partial x^{2}} +\frac{%
\partial ^{2}u}{\partial y^{2}}=f(x,y) \qquad \qquad $Laplace/Poisson

\item $x\frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial
y^{2}}=0$ \qquad \qquad Tricomi

\item $\frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial
y^{2}}+k^{2}u=f(x,y)$ \qquad \qquad Helmholtz

\item $\frac{\partial ^{2}u}{\partial t^{2}}-\frac{\partial ^{2}u}{\partial
x^{2}}=0$ \qquad \qquad Wave

\item $\frac{\partial u}{\partial t}-\kappa \frac{\partial ^{2}u}{\partial
x^{2}}=0$ \qquad \qquad heat/diffusion \ \ \ \ \ $\kappa >0$

\item $\frac{\partial ^{2}u}{\partial t^{2}}+\kappa \frac{\partial ^{4}u}{%
\partial x^{4}}=0$ \qquad \qquad vibrating bar \quad $\kappa >0$

\item $\frac{\partial u}{\partial t}-i\frac{\partial ^{2}u}{\partial x^{2}}%
=0 $ Schroedinger

\item $(2\alpha +\beta )\frac{\partial ^{2}H}{\partial x^{2}}+(\alpha
+2\beta )\frac{\partial ^{2}H}{\partial y^{2}}=0$ \ \ \ glass manufacture $%
u=\alpha x\qquad v=\beta y$

\item $u=\frac{\partial \varphi }{\partial x}\qquad v =\frac{\partial \varphi%
}{\partial y}$ \qquad $(a^{2}-u^{2})\frac{\partial ^{2}\varphi }{\partial
x^{2}}-2uv\frac{\partial ^{2}\varphi }{\partial x\partial y}+(a^{2}-v^{2})%
\frac{\partial ^{2}\varphi }{\partial y^{2}}=0$ \ \ \ \ potential

\item System

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

$\frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}=0$ \qquad
\qquad Cauchy-Riemann

\item $\frac{\partial u}{\partial t}-c\frac{\partial v}{\partial x}=0$

$\frac{\partial v}{\partial t}-c\frac{\partial u}{\partial x}=0$ \ 

\item $\frac{\partial p}{\partial t}-\frac{\partial u}{\partial x}-\frac{%
\partial v}{\partial y}=0$

$\frac{\partial u}{\partial t}-\frac{\partial p}{\partial x}=0$

$\frac{\partial v}{\partial t}-\frac{\partial p}{\partial y}=0$ \qquad
\qquad acoustics

\item $\frac{\partial p}{\partial t}+u\frac{\partial p}{\partial x}+v\frac{%
\partial p}{\partial y}+\gamma p(\frac{\partial u}{\partial x}+\frac{%
\partial v}{\partial y})=0$

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{%
\partial u}{\partial y}+\frac{1}{\rho }\frac{\partial p}{\partial x}=0$

$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{%
\partial v}{\partial y}+\frac{1}{\rho }\frac{\partial p}{\partial y}=0$
\qquad Euler

\item $\frac{\partial B}{\partial t}+c\quad curl(E)=0$

$\frac{\partial E}{\partial t}-c\quad curl(B)=-4\pi J$ \qquad Maxwell
\end{itemize}

\begin{center}
Well-posedness
\end{center}

\begin{itemize}
\item existence

\item uniqueness

\item continuous dependence
\end{itemize}

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