MIT OpenCourseWarehttp://ocw.mit.edu 18.306 Advanced Partial Differential Equations with ApplicationsFall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 08 2009 10 05 MON TOPICS: More on envelopes. Infinite slopes at envelope. Shocks. Conservation and entropy. Irreversibility. Examples from traffic flow. Continue with c_t + c*c_x = 0 and c(x, 0) = C(x). Show alternative definition of envelope of a smooth family of curves: Curve such that each point belongs to a family member, and is tangent to the member here. Hence: characteristics are tangent to the boundary of multiple values. Generic drawing of multiple valued region now justified. Back to conservation form: \rho_t + q_x = 0. Introduce shocks to knock out multiple-valued regions. Now pde + discontinuities satisfying some conditions: Rankine-Hugoniot jump conditions (conservation) Lax entropy conditions (causality) System is now IRREVERSIBLE (show how information is lost at shocks). Simple examples in Traffic Flow: Red light turns green (show how Lax Entropy crucial here). Green light turns red. Green-red Red-green Meaning and (qualitative) comparison with reality. Generic prescription for shocks forming out of a smooth
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