Editing Method of characteristics (section)

===Linear and quasilinear cases===
Consider now a PDE of the form

:<math>\sum_{i=1}^n a_i(x_1,\dots,x_n,u) \frac{\partial u}{\partial x_i}=c(x_1,\dots,x_n,u).</math>

For this PDE to be [[linear]], the coefficients ''a''<sub>''i''</sub> may be functions of the spatial variables only, and independent of ''u''.  For it to be quasilinear,<ref name="quasilinear">{{cite web| url = https://reference.wolfram.com/language/tutorial/DSolveLinearAndQuasiLinearFirstOrderPDEs.html |title = Partial Differential Equations (PDEs)—Wolfram Language Documentation}}</ref> ''a''<sub>''i''</sub> may also depend on the value of the function, but not on any derivatives.  The distinction between these two cases is inessential for the discussion here.

For a linear or quasilinear PDE, the characteristic curves are given parametrically by

:<math>(x_1,\dots,x_n,u) = (X_1(s),\dots,X_n(s),U(s))</math>
:<math>u(\mathbf{X}(s)) = U(s)</math>
for some univariate functions <math>s\mapsto (X_i(s))_i,U(s)</math>
of one real variable <math>s</math>
satisfying the following system of ordinary differential equations

{{NumBlk|:|<math>X_i' = a_i(X_1,\dots,X_n,U)
\text{ for }i=1,\dotsc,n</math>|{{EquationRef|2}}}}
{{NumBlk|:|<math>U' = c(X_1,\dots,X_n,U).</math>|{{EquationRef|3}}}}

Equations ({{EquationNote|2}}) and ({{EquationNote|3}}) give the characteristics of the PDE.

{{collapse top|title=Proof for quasilinear case}} 
In the quasilinear case, the use of the method of characteristics is justified by [[Grönwall's inequality]]. The above equation may be written as
<math display="block">\mathbf{a}(\mathbf{x},u) \cdot \nabla u(\mathbf{x}) = c(\mathbf{x},u) </math>

We must distinguish between the solutions to the ODE and the solutions to the PDE, which we do not know are equal ''a priori.'' Letting capital letters be the solutions to the ODE we find
<math display="block">\mathbf{X}'(s) = \mathbf{a}(\mathbf{X}(s),U(s)) </math>
<math display="block">U'(s) = c(\mathbf{X}(s), U(s)) </math>

Examining <math>\Delta(s) = |u(\mathbf{X}(s)) - U(s)|^2 </math>, we find, upon differentiating that
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\mathbf{X}'(s)\cdot \nabla u(\mathbf{X}(s)) - U'(s)\Big) </math>
which is the same as
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\mathbf{a}(\mathbf{X}(s),U(s))\cdot \nabla u(\mathbf{X}(s)) - c(\mathbf{X}(s),U(s))\Big) </math>

We cannot conclude the above is 0 as we would like, since the PDE only guarantees us that this relationship is satisfied for
<math>u(\mathbf{x})</math>, <math>\mathbf{a}(\mathbf{x},u) \cdot \nabla u(\mathbf{x}) = c(\mathbf{x},u)</math>,
and we do not yet know that <math>U(s) = u(\mathbf{X}(s))</math>.

However, we can see that
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\mathbf{a}(\mathbf{X}(s),U(s))\cdot \nabla u(\mathbf{X}(s)) - c(\mathbf{X}(s),U(s))-\big(\mathbf{a}(\mathbf{X}(s),u(\mathbf{X}(s))) \cdot \nabla u(\mathbf{X}(s)) - c(\mathbf{X}(s),u(\mathbf{X}(s)))\big)\Big) </math>
since by the PDE, the last term is 0. This equals
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\big(\mathbf{a}(\mathbf{X}(s),U(s))-\mathbf{a}(\mathbf{X}(s),u(\mathbf{X}(s)))\big)\cdot \nabla u(\mathbf{X}(s)) - \big(c(\mathbf{X}(s),U(s))-c(\mathbf{X}(s),u(\mathbf{X}(s)))\big)\Big) </math>

By the triangle inequality, we have
<math display="block">|\Delta'(s)| \leq 2\big|u(\mathbf{X}(s)) - U(s)\big| \Big(\big\|\mathbf{a}(\mathbf{X}(s),U(s))-\mathbf{a}(\mathbf{X}(s),u(\mathbf{X}(s)))\big\| \ \|\nabla u(\mathbf{X}(s))\| + \big|c(\mathbf{X}(s),U(s))-c(\mathbf{X}(s),u(\mathbf{X}(s)))\big|\Big) </math>

Assuming <math>\mathbf{a},c </math> are at least <math>C^1 </math>, we can bound this for small times. Choose a neighborhood <math>\Omega </math> around <math>\mathbf{X}(0), U(0) </math> small enough such that <math>\mathbf{a},c </math> are [[locally Lipschitz]]. By continuity, <math>(\mathbf{X}(s),U(s)) </math> will remain in <math>\Omega </math> for small enough <math>s
 </math>. Since <math>U(0) = u(\mathbf{X}(0)) </math>, we also have that <math>(\mathbf{X}(s), u(\mathbf{X}(s))) </math> will be in <math>\Omega </math> for small enough <math>s </math> by continuity. So, <math>(\mathbf{X}(s),U(s)) \in \Omega </math> and <math>(\mathbf{X}(s), u(\mathbf{X}(s))) \in \Omega </math> for <math>s \in [0,s_0] </math>. Additionally, <math>\|\nabla u(\mathbf{X}(s))\| \leq M </math> for some <math>M \in \R </math> for <math>s \in [0,s_0] </math> by compactness. From this, we find the above is bounded as
<math display="block">|\Delta'(s)| \leq C|u(\mathbf{X}(s)) - U(s)|^2 = C |\Delta(s)| </math>
for some <math>C \in \mathbb{R} </math>. It is a straightforward application of Grönwall's Inequality to show that since <math>\Delta(0) = 0 </math> we have <math>\Delta(s) = 0 </math> for as long as this inequality holds. We have some interval <math>[0, \varepsilon) </math> such that <math>u(X(s)) = U(s) </math> in this interval. Choose the largest <math>\varepsilon </math> such that this is true. Then, by continuity, <math>U(\varepsilon) = u(\mathbf{X}(\varepsilon)) </math>. Provided the ODE still has a solution in some interval after <math>\varepsilon </math>, we can repeat the argument above to find that <math>u(X(s)) = U(s) </math> in a larger interval. Thus, so long as the ODE has a solution, we have <math>u(X(s)) = U(s) </math>.
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