2 - First Order Scalar Quasilinear Equations

# 2 - First Order Scalar Quasilinear Equations - MATH 220...

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MATH 220: FIRST ORDER SCALAR QUASILINEAR EQUATIONS We now consider the quasilinear equations; these have the form (1) a ( x, y, u ) u x + b ( x, y, u ) u y = c ( x, y, u ) , with a, b, c at least C 1 , given real valued functions. There is an immediate diFerence between semilinear and quasilinear equations at this point: since a and b depend on u , we cannot associate a vector ±eld on R 2 to the equation: we need to work on R 3 at least to account for the ( x, y, u ) dependence. To achieve this, we proceed as follows. We consider the graph S of u in R 2 × R , given by z = u ( x, y ). If we know the solution u , we know the graph, and conversely we can recover u if we ±nd its graph, a surface in R 3 . So let U ( x, y, z ) = u ( x, y ) z ; we need to ±nd the 0-set of U . Now, substituting U into the left hand side of the PDE we get ( a ( x, y, u ( x, y )) x + b ( x, y, u ( x, y )) y ) U ( x, y, z ) = a ( x, y, u ( x, y )) u x ( x, y ) + b ( x, y, u ( x, y )) u y ( x, y ) . If the PDE holds, this equals c ( x, y, u ( x, y )), which is in turn equal to c ( x, y, u ( x, y )) z U as z U ≡ − 1! Thus, our PDE implies that ( a ( x, y, u ( x, y )) x + b ( x, y, u ( x, y )) y + c ( x, y, u ( x, y )) z ) U ( x, y, z ) = 0 for all x, y, z . This equation states that the directional derivative of U along the vector ±eld ( a, b, c ) vanishes. Note that z U = 1, i.e. never vanishes, so in fact at each point ( x, y, z ) the directional derivative can only vanish along a plane (and not in every direction). But along any level set of U its directional derivative certainly vanishes, so applied to the 0-level set, where u ( x, y ) = z , the PDE states that the vector ±eld W ( x, y, z ) = ( a ( x, y, z ) , b ( x, y, z ) , c ( x, y, z )) is tangent to S . We can now rephrase our problem as follows: we want to ±nd S as a union of integral curves of W . As before, integral curves of W , where we introduce a parameter r again, satisfy ∂x ∂s ( r, s ) = a ( x ( r, s ) , y ( r, s ) , z ( r, s )) , ∂y ∂s ( r, s ) = b ( x ( r, s ) , y ( r, s ) , z ( r, s )) , ∂z ∂s ( r, s ) = c ( x ( r, s ) , y ( r, s ) , z ( r, s )); these are called the characteristic ODEs and integral curves are called the charac- teristics , though perhaps it would be most appropriate to call them lifted character- istics in view of our terminology for semilinear equations. Note that although the integral curves are in R 3 , hence there is a two-dimensional family of them, we only care about the ones which will give S as their union, i.e. a one-dimensional family 1

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2 of integral curves. Our initial condition will again come from the initial condition of the PDE, which is of the form u (Γ( r )) = φ ( r ) along a curve Γ( r ) = (Γ 1 ( r ) , Γ 2 ( r )). The initial condition states that over the curve Γ, S is given by z = φ ( r ), i.e. that S goes through the curve ˜ Γ( r ) = (Γ 1 ( r ) , Γ 2 ( r ) , φ ( r )) . Our initial condition for the characteristic ODEs is then
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2 - First Order Scalar Quasilinear Equations - MATH 220...

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