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pde1-2

# pde1-2 - PDE LECTURE NOTES MATH 237A-B BRUCE K DRIVER...

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PDE LECTURE NOTES, MATH 237A-B BRUCE K. DRIVER Abstract. These are lecture notes from Math 237A-B. See C: \ driverdat \ Bruce \ CLASSFIL \ 257AF94 \ course.tex for notes on contraction semi-groups. Need to add examples of using the Hille Yoshida theorem in PDE. See See “C: \ driverdat \ Bruce \ DATA \ MATHFILE \ qft-notes \ co-area.tex” for co- area material and applications to Sobolev inequalities. Contents 1. Some Examples 1 1.1. Some More Geometric Examples 5 2. First Order Quasi-Linear Scalar PDE 6 2.1. Linear Evolution Equations 6 2.2. General Linear First Order PDE 13 2.3. Quasi-Linear Equations 19 2.4. Distribution Solutions for Conservation Laws 23 2.5. Exercises 27 3. Fully nonlinear fi rst order PDE 32 3.1. An Introduction to Hamilton Jacobi Equations 35 4. Cauchy — Kovaleski Theorem 43 5. Test Functions and Partitions of Unity 46 5.1. Convolution: 47 6. Cuto ff functions and Partitions of Unity 48 7. Surfaces and Surface Integrals 50 8. Laplace’s and Poisson’s Equation 59 9. Laplacian in polar coordinates 64 9.1. Laplace’s Equation Poisson Equation 64 10. Estimates on Harmonic functions 79 11. Green’s Functions 81 12. A Little Distribution Theory 87 12.1. Max Principle a la Hamilton 99 13. Wave Equation 100 13.1. Corresponding fi rst order O.D.E. 102 13.2. Spherical Means 111 14. Old Section Stu ff 118 Date : October 23, 2002 File:zpde.tex . Department of Mathematics, 0112. University of California, San Diego . La Jolla, CA 92093-0112 . i

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ii BRUCE K. DRIVER 14.1. Section 2 118 14.2. Section 3 120 15. Solutions to Exercises 121 15.1. Section 2 Solutions 121
PDE LECTURE NOTES, MATH 237A-B 1 1. Some Examples Example 1.1 (Tra c Equation) . Consider cars travelling on a straight road, i.e. R and let u ( t, x ) denote the density of cars on the road at time t and space x and v ( t, x ) be the velocity of the cars at ( t, x ) . Then for J = [ a, b ] R , N J ( t ) := R b a u ( t, x ) dx is the number of cars in the set J at time t. We must have Z b a ˙ u ( t, x ) dx = ˙ N J ( t ) = u ( t, a ) v ( t, a ) u ( t, b ) v ( t, b ) = Z b a ∂x [ u ( t, x ) v ( t, x )] dx. Since this holds for all intervals [ a, b ] , we must have ˙ u ( t, x ) dx = ∂x [ u ( t, x ) v ( t, x )] . To make life more interesting, we may imagine that v ( t, x ) = F ( u ( t, x ) , u x ( t, x )) , in which case we get an equation of the form ∂t u = ∂x G ( u, u x ) where G ( u, u x ) = u ( t, x ) F ( u ( t, x ) , u x ( t, x )) . A simple model might be that there is a constant maximum speed, v m and maxi- mum density u m , and the tra c interpolates linearly between 0 (when u = u m ) to v m when ( u = 0) , i.e. v = v m (1 u/u m ) in which case we get ∂t u = v m ∂x ( u (1 u/u m )) . Example 1.2 (Burger’s Equation) . Suppose we have a stream of particles travelling on R , each of which has its own constant velocity and let u ( t, x ) denote the velocity of the particle at x at time t. Let x ( t ) denote the trajectory of the particle which is at x 0 at time t 0 . We have C = ˙ x ( t ) = u ( t, x ( t )) . Di ff erentiating this equation in t at t = t 0 implies 0 = [ u t ( t, x ( t )) + u x ( t, x ( t )) ˙ x ( t )] | t = t 0 = u t ( t 0 , x 0 ) + u x ( t 0 , x 0 ) u ( t 0 , x 0 ) which leads to Burger’s equation 0 = u t + u u x .

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pde1-2 - PDE LECTURE NOTES MATH 237A-B BRUCE K DRIVER...

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