Unformatted text preview: Problem 6. Consider the PDE u t + uu x = 0 , u ( x, 0) = φ ( x ) . Suppose that φ ′ ≥ C , where C > 0. Show that the PDE has a C 1 solution on R x × [0 , 1 C ) t . Show also that for t ∈ [0 , 1 C ), u x satis±es the estimate u x ( x, t ) ≥ 1 tC − 1 . (Note that the right hand side is negative!) (Hint: Consider the diFerence quotients u ( ξ 2 ( t ) ,t ) − u ( ξ 1 ( t ) ,t ) ξ 2 ( t ) − ξ 1 ( t ) , where x = ξ j ( t ) are the projected characteristic curves emanating from the point x j on the x axis.) 1...
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 '09
 Math, Differential Equations, Applied Mathematics, Equations, Derivative, Partial Differential Equations, Complex number, Nonlinear system, PDEs, Solve ux

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