Applied PDEs: Analysis and Computation

Transcription

1 Applied PDEs: Analysis and Computation Hailiang Liu Iowa State University Tsinghua University May 07 June 16, / 15

2 Lecture #1: Introduction May 09, 2012 Model, Estimate and Algorithm=MEA M : 2, A showcase of classical PDEs E : 1 < 2 < 2, A : x n+1 = xn x n. PDE arose in the context of the development of models in the physics of continuous media, e.g. vibrating strings, elasticity, the Newtonian gravitational field of extended matter, electrostatics, fluid flows, and later by the theories of heat conduction, electricity and magnetism. In addition, problems in differential geometry gave rise to nonlinear PDEs such as the Monge-Ampere equation and the minimal surface equations. - Three classical PDEs by Euler, Laplace and Fourier - PDEs from 1752 to PDEs after 1900 Choice of research topics and main issues of interest 2 / 15

3 Turning points of the PDE s theory development Hilbert s 19th problem: whether the solutions of regular problems in the calculus of variations are always analytic. 1851: Riemann, tried to prove existence using variational formulation 1870: Weierstrass rigorization program began the era of PDE theory. 1870: Poincare gave first complete proof for Laplace equation 1875: Kowalevska (PDE); 1842, Cauchy (ODE), local analytic solution using power series 1898: Poincare s continuity method 1900: Hilbert s 19th problem 1906: Bernstein, the beginning of a priori estimates 1920: the concept of weak solutions using distributions 1923: Hadamard: the concept of well-posedness 1930: Sobolev spaces 1934: Regularity Morrey 2D; 1957: DeGiorgi and J. Nash, summit of the regularity achievement The need to develop a rigorous PDE theory was a strong motivation in the development of basic tools in real analysis and functional analysis since the beginning of 20th century. 3 / 15

4 Lecture #2: First order PDEs May 11, 2012 F (x, u, u) = 0. The classical calculus of variations in the form of the Euler-Lagrange principle gave rise to PDEs and the Hamilton-Jacobi theory, which had arisen in mechanics, stimulated the analysis of first order PDEs. Applications: mechanics, computer vision, image processing, geometry based motion. Local solution by the method of characteristics (Classical) Multi-vlaued solution by the Level set method (Liu-Cheng-Osher, 2003 ) Let φ be an implicit function in jet space (x, z, p) such that where φ is obtained by solving (u, u) {(z, p), φ(x, z, p) = 0}, F p x φ + p F p z φ (F x + zf p) p = 0. Ref. Liu, Hailiang and Cheng, Li-Tien and Osher, Stanley. Level set framework for capturing multi-valued solutions of nonlinear first order equations. J. Sci. Comput. 29: , Multi-vlaued solution: Geometric optics in high frequency wave propagation 4 / 15

5 Lecture #3: Viscosity vs entropy solution May 16, 2012 Given data for a PDE problem, either the solution gets better or worse. For the former, some coercivity is essential for the regularity estimate. For the later, one has to come up with appropriate weak solutions so that the solution remains unique (such as the viscosity solution and the entropy solution). More applications: game theory, optimal control, stochastic control... *Viscosity solution for Hamilton-Jacobi equations (Crandall-Lions, 1983 ) Evans(1980), Ishii(1992), Barles(1994), Giga(1991), Caffarelli(1995). Ref: M. Crandall, H. Ishii and P. L. Lions, Users guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), *Entropy solution for scalar conservation laws (Kruzkov, 1970 ) u 0 L 1 loc, u CtL1 loc for ut + x F (u) = 0. Lax(1957),Oleinik(1963), Volpert (1967), Kuznetsov (1976) Ref: F. Bouchu and B Perthame. Kruzkovs estimates for scalar conservation laws revisited. Transactions of the American Mathematical Society, 350(7), 1998, / 15

6 Lecture #4: Second order and ellipticity May 18, 2012 We consider an elliptic operator n Lu = i (a ij j u), ρ(a ij ) > 0. i=1 Three fundamental equations - Hyperbolic wave equation - Parabolic equation - Elliptic equation u tt Lu = 0. u t = Lu. Lu = f. Energy method for wave equation E(h) = [ut 2 + a ij u xi u xj ]dx, R(h) where R(h) is the horizontal part at time h within the backward characteristic cone from (x, t). Theorem If u = u t = 0 on R(0), then u = 0 within the characteristic cone. 6 / 15

7 Lecture #5: Ellipticity and regularity May 23, 2012 Let the elliptic operator be Lu = n i=1 i (a ij j u), ρ(a ij ) > 0. The basic result in the theory of weak solutions for PDEs is the Sobolev imbedding theorem. - If 1 p < n, then u W 1,p implies u L p with p = np n p - If p > n, W 1,p is Hölder continuous. u p C(n, p) Du p. [u] α C(n, p) Du p. DeGiorgi s estimate: If u L p and f L p with p > n/2, then u L for Lu = f. Nash s estimate for u t = Lu, u(x, 0) = δ(x). 7 / 15

8 Lecture #6: Regularity estimate May 25, 2012 Continuous version of DeGiorgi s estimate - u L p then u L p +1 - v = (u s) +, F (s) = v p +1 - Derive Ḟ CF β, β < 1. Nash s estimate for L 1 initial data: - Energy estimate Nash s inequality - Point-wise estimate (semi-group) d dt E + cm u kt n/2. u 2 dx 0. 8 / 15

9 Lecture #7: Entropy and moment bound May 30, 2012 Moment bound - Nash entropy approach using Q = ulogu to prove C 1 x udx t C. Entropy satisfying methods - Why kinetic equations? - Entropy structure - Choice of discretization Two applications: 1. Polymer dynamics: 2. Biological dispersal f t = m ( mf + Ff umf ), F = bm b m 2. f t = x ( x f + x Uf ) + λu(m(x) u). 9 / 15

10 Lecture #8: Entropy and free energy June 01, 2012 Boltzmann equation - H function and H theorem Fokker-Planck equation - Free energy H = E = f logf f logf + Uf Non-local Fokker-Planck equation - Free energy E = f logf Uf, U = W f. Convergence towards equilibrium f f 1 Ce λt. 10 / 15

11 CAM seminar: analysis and computation June 02, 2012 Math and physics; central problem? (Einstein and Peter Lax) Analysis vs computation: - Von Neumann s program - Numerical discovery of solitons, Kruskal and Zabusky My research: Numerical Modeling: level set, recovery Threshold analysis: polymer dynamics, photon transport, biological dispersal phase transition, Critical threshold Computation methods: AE, DDG, entropy satisfying Historical figures: - Galileo, Kepler (telescope, logrithmic) - Newton, Euler: - Gauss, Riemann - Lagrange, Laplace, Fourier, Poisson, Cauchy, Liouville - Green, Airy, Stokes, Thomson, Hamilton - Rayleigh, Maxwell - Boltzmann and Gibbs; Schwartz, Onsager - Einstein - Weyl, Von Neuman, Friedrichs, Kato Dyson, Schroedinger 11 / 15

12 Lecture #9: Entropy vs invariant region June 6, 2012 Boltzmann equation - H function and H theorem H = f logf Two useful inequalities - CPK inequality - Log-Sobolev inequality h 2 logh 2 dµ c f f 1 2 h 2 dµ + f log f f. h 2 dµlog( h 2 dµ). Invariant region for reaction-diffusion-convection systems (one dimensional). This concept extends the usual maximum principle for scalar equations. The invariant region may be an open set, the bound on the open side may be obtained using the energy method. 12 / 15

13 Lecture #10: Energy method June 8, 2012 Advantages of the energy method Range of applicability - Artificial viscosity where ε > 0 and p < 0, p > 0. - Gradient systems v t u x = εv xx, u t + p(v) x = εu xx, u t + (φ u) x = Bu xx Here φ(0) = 0, and φ grows slower than u 4. - Isentropic Gas dynamics { vt u x = 0, u t + p(v) x = (k(v)u x ) x, where p < 0, p > 0, and k(v) > 0 for v > 0. - Incompressible Navier-Stokes equation tu + (u x )u = x p + u, u = / 15

14 Lecture #11: Numerical methods June 13, 2012 How to compute PDE solutions discussed so far? 3R pre-formulation - Variation of formulations - Model refinement: flux refinement - Model enhancement: level set method for computing multi-valued solutions - Model reduction: Geometric optics, recovery Choice of discretization - FXMs= Finite { Difference, Volume, Element } methods. - FDM: L -norm based; sample at grid nodes. - FVM: L 1 -norm based; sample through cell averages and reconstruction - FEM: L 2 -norm based; sample through moments 14 / 15

15 Lecture #12: DG methods and beyond June 15, 2012 Alternating evolution - Hamilton-Jacobi equation, fully nonlinear equation DDG with flux refinement - PDEs in certain conservative form: Hyperbolic conservations, diffusion equations Entropy satisfying - Fokker-Planck equations. 15 / 15