$\newcommand{\const}{\mathrm{const}}$

## First order PDEs

#### Introduction

Consider PDE $$au_t+bu_x=0.$$ Note that the left-hand expression is a directional derivative of $u$ in the direction $\ell=(a,b)$. Consider an integral lines of this vector field: $$\frac{dt}{a}=\frac{dx}{b}.$$ Remark. Recall from ODE cours that an integral line of the vector field is a line, tangent to it in each point.

#### Constant coefficients

If $a$ and $b$ are constant then integral curves are just straight lines $t/a -x/b=C$ where $C$ is a constant along integral curves and it labels them (at least as long as we consider the whole plane $(x,t)$). Therefore $u$ depends only on $C$: $$u= \phi \bigl( \frac{t}{a}-\frac{x}{b}\bigr)$$ wher $\phi$ is an arbitrary function.

This is a general solution of our equation.

Consider initial value condition $u|_t=0=f(x)$. It allows us define $\phi$: $\phi(x)=f(x)$. Plugging in $u$ we get $$u=f\bigl( \frac{t}{a}-\frac{x}{b}\bigr).$$ It is a solution of IVP \left\{\begin{aligned} &au_t+bu_x=0.\\ &u(x,0)=f(x). \end{aligned} \right. Obviously we need to assume that $a\ne 0$.

If $a=1$ we can rewrite general solution in the form $u(x,t)=\phi_1 (x-bt)$ where $\phi_1(x)=\phi(-xb^{-})$ is another arbitrary function.

Definition Solutions $u= \chi(x-ct)$ are running waves where $c$ is a propagation speed.

#### Variable coefficients

If $a$ and/or $b$ are not constant these integral lines are curves.

Example. Consider equation $u_t+tu_x=0$. Then equation of the integral curve is $\frac{dt}{1}=\frac{dx}{t}$ or equivalently $tdt-dx=0$ which solves as $x-\frac{1}{2}t^2=C$ and therefore $u=\phi (x-\frac{1}{2}t^2)$ is a general solution to this equation.

One can see easily that $u=f((x-\frac{1}{2}t^2)$ is a solution of IVP.

Example. Consider the same equation but let us consider IVP as $x=0$: $u(0,t)=g(tx)$. However it is not a good problem: first, some integral curves intersect line $x=0$ more than once and if in different points of intersection of the same curve initial values are different we get a contradiction (therefore problem is not solvable for $g$ which are not even functions).

On the other hand, if we consider even function $g$ (or equivalently impose initial condition only for $t>0$) then $u$ is not defined on the curves which are not intersecting $x=0$ (which means that $u$ is not defined for $x>\frac{1}{2}t^2$.)

In this example both solvability and unicity are broken.

#### Right-hand expression

Consider equation $$au_t+bu_x=f.$$ Then as $\frac{dt}{a}=\frac{dx}{b}$ we have $du = u_t dt + u_xdx = (au_t+bu_x) \frac{dt}{a}=f \frac{dt}{a}$ and therefore we expand our ordinary equation to $$\frac{dt}{a}=\frac{dx}{b}=\frac{du}{f}.$$

#### Linear and semilinear equations

Definition If $a=a(x,t)$and $b=b(x,t)$ equation is semilinear.

In this case we first define integral curves which do not depend on $u$ and then find $u$ as a solution of ODE along these curves.

Definition Furthermore if $f$ is a linear function of $u$: $f=c(x,t)u + g(x,t)$ original equation is linear.

In this case the last ODE is also linear.

Example. Consider $u_t+ xu_x = x t$. Then $\frac{dt}{1}=\frac{dx}{x}=\frac{du}{tu}$. Solving the first equation $t-\ln x=-\ln C\implies x =Ce^t$ we get integral curves. Now we have \begin{multline*} \frac{du}{xt}=dt \implies du= x t dt= Cte^t dt \implies u=C(t-1)e^t +C_1 =\\ x(t-1)+C_1\end{multline*} where $C_1$ must be constant along integral curves and therefore $C_1=\phi (xe^{-t})$ with an arbitrary function $\phi$. So $u=x(t-1)+\phi (xe^{-t})$ is a general solution of this equation.

Imposing initial condition $u|_{t=0}=0$ (sure, we could impose another condition) we have $\phi(x)=x$ and then $u=x(t-1 +e^{-t})$.

#### Quasilinear equations

Definition. If $a$ and/or $b$ depend on $u$ this is quasininear equation.

For such equations integral curves depend on the solution which can lead to breaking of solution.

Example. Consider Hopf equation $u_t+uu_x=0$ (which is an extremely simplified model of gas dynamics. ) We have $\frac{dt}{1}=\frac{dx}{u}=\frac{du}{0}$ and therefore $u=\const$ along integral curves and therefore integral curves are $x-ut=C$.

Consider initial problem $u(x,0)=f(x)$. We take initial point $(y,0)$, find here $u=f(y)$, then $x-f(y)t =y$ (think why?) and we get $u=f(y)$ where $y=y(x,t)$ is a solution of equation $x=f(y)t +y$.

The trouble is that we can define $y$ for all $x$ only if $\frac{\partial }{\partial y}\bigl(f(y)t +y\bigr)$ does not vanish. So, $f'(y)t +1\ne 0$.

This is possible for all $t>0$ if and only if $f'(y)\ge 0$ i.e. $f$ is a monotone non-decreasing function.

So, classical solution breaks if $f$ is not a monotone non-decreasing function. A proper understanding of the global solution for such equation goes well beyond our course.

#### IBVP

Consider IVP for constant coefficient equation \left\{\begin{aligned} &u_t +cu_x=0, \qquad &&x>0, t>0,\\ &u|_{t=0}= f(x) \qquad &&x>0. \end{aligned}\right. The general solution is $u=\phi(x-ct)$ and plugging into initial data we get $\phi(x)=f(x)$ (as $x>0$).

So, $u(x,t)= f(x-ct)$. Done!–Not so fast. $f$ is defined only for $x>0$ so $u$ is defined for $x-ct>0$ (or $x>ct$). It covers the whole quadrant if $c\le 0$ (so waves run to the left) and only in this case we are done.

If $c>0$ (waves run to the right) $u$ is not defined as $x< ct$ and to define it here we need a boundary condition at $x=0$. So we get IBVP (initial-boundary value problem) \left\{\begin{aligned} &u_t +cu_x=0, \qquad &&x>0, t>0,\\ &u|_{t=0}= f(x) \qquad &&x>0,\\ &u|_{x=0}=g(t) \qquad &&t>0. \end{aligned}\right.

Then we get $\phi(-ct)=g(t)$ as $t>0$ which implies $\phi(x)=g(-\frac{1}{c}x)$ as $x<0$ and then $u(x,t)=g(-\frac{1}{c}(x-ct))=g(t-\frac{1}{c}x)$ as $x< ct$.

So solution is u=\left\{\begin{aligned} &f(x-ct)\qquad &&x> c t,\\ &g(t-\frac{1}{c}x)\qquad && x < ct. \end{aligned}\right.

Remark. Nonlinear equation $$F(x_1,x_2,u,u_{x_1},u_{x_2})=0$$ (we prefer such notations here) also could be solved through ODEs but it is much more complicated: one needs simultaneously find $x,u$ and $p_j=u_{x_j}$ along trajectories from the system of equations: \left\{\begin{aligned} &\frac{dx_j}{dt}=F_{p_j}, \\ &\frac{dp_j}{dt}=-F_{x_j}-F_u p_j ,\\ &\frac{du}{dt}=\sum_{j=1}^n F_{p_j}p_j \end{aligned}\right. where in the right hand expressions we consider as function of $2n+1$ variables $x_1,\ldots,x_n, u, p_1,\ldots, p_n$, $n=2$.
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#### Multidimensional equations

Remark. Multidimensional equations (from linear to semilinear) $$au_t + \sum_{j=1}^n b_j u_{x_j}=f(x_1,\ldots,x_n,t,u)$$ and nonlinear $$F(x_1,\ldots,x_n,t,u,u_{x_1},\ldots,u_{x_n},u_t)=0$$ could be solved by the same methods.

For example, if $a=1$, $b_j=\const$ and $f=0$ the general solution is $u=\phi (x_1-b_1t,\ldots,x_n-b_nt)$ where $\phi$ is an arbitrary function of $n$ variables.