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**Unformatted text preview: **Problem 6. Consider the PDE u t + uu x = 0 , u ( x, 0) = ( x ) . Suppose that -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 satises the estimate u x ( x, t ) 1 t-C 1 . (Note that the right hand side is negative!) (Hint: Consider the diFerence quotients u ( 2 ( t ) ,t ) u ( 1 ( t ) ,t ) 2 ( t ) 1 ( t ) , where x = j ( t ) are the projected characteristic curves emanating from the point x j on the x axis.) 1...

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