Struwe, the cauchy problem for wave maps, international mathematics research notices, no. We explain how to apply techniques from integrable sy. Struwe, well posedness in energy space for semilinear wave. Shatah and struwe, geometric wave equations keel and tao, endpoint strichartz estimatesin american journal of mathematics folland, real analysis. Analysis and geometry volume 11, number 1, 4983, 2003 on the wellposedness of the wave map problem in high dimensions andrea nahmod, atanas stefanov and karen uhlenbeck we construct a gauge theoretic change of variables for the wave. Struwe, wellposedness in the energy space for semilinear wave equations with critical growth, imrn, 7. New york university, courant institute of mathematical sciences, new york. Existence and uniqueness of global so lutions is known for the wave equations for an arbitrary target manifold provided. We prove the uniqueness of weak solutions to the critical defocusing wave equation in 3d under a local energy inequality condition. Bibliography seminar on wave equations and black hole. Exact solutions are derived for an ndimensional radial wave equation with a general power nonlinearity. Geometric wave equations courant lecture notes 9780821827499. Blow up of the solutions of nonlinear wave equation in. Uniform decay estimates and the lorentz invariance of the classical wave equation, 1985.
Zeng, geometry and a priori estimates for free boundary problems of the euler equation, communications on pure and. Both our work and that of shatah struwe 11 use the same gauge change. A standing wave, also known as a stationary wave, is a wave that remains in a constant position. In 8, the energy ew, 0 has been shown to be a threshold for the dynamical behavior of solutions of the equation.
Blow up of the solutions of nonlinear wave equation. On the lifespan of nonlinear wave equations with small initial data, 1990. New york university courant institute of mathematical sciences, 1998, vol. Besov spaces are contained in lorentz spaces for appropriate indeces c. For more detailed results on wave map, one can refer to the book of shatah and struwe.
The method, which is applicable more generally to other nonlinear pdes, involves an ansatz technique to solve a firstorder pde system of groupinvariant variables given by group foliations of the wave equation, using the onedimensional admitted point symmetry groups. Geometric wave equations by jalal m ihsan shatah, 9780821827499, available at book depository with free delivery worldwide. Physical theory local existence nonlinear wave equation isometric embedding. Dynamics of threshold solutions for energycritical wave. This volume contains notes of the lectures given at the courant institute and a dmvseminar at oberwolfach. Under some assumptions on the nonlinearities, it was shown that uniqueness of. Geometric wave equations, courant lecture notes in mathematics. Gilbarg and trudinger, elliptic partial differential equations of second order. Constrained wave equations and wave maps springerlink.
In this paper we establish that wave maps can be obtained by a penalization method if the initial data is well prepared. Andrew lawrie joint with sungjin oh and sohrab shahshahani wave maps from the hyperbolic plane. A supercritical nonlinear wave equation in 2 space dimensions. Geometric wave equations courant lecture notes by jalal. Stable soliton resolution for equivariant wave maps exterior. Here we answer of the open problem 0 using approach. Moulik kallupalam balasubramanian rutgers university. Some results for the daveystewartson system on a circle. On the wellposedness of the wave map problem in high. By continuing to use our website, you are agreeing to our use of cookies. The book is suitable for use in a graduatelevel course on. Pdf on uniqueness for supercritical nonlinear wave and. Struwe, regularity results for nonlinear wave equations, ann.
Regularity results for semilinear and geometric wave equations. Existence and uniqueness of global solutions is known for the wave equations for an arbitrary target manifold provided. Global wellposedness and scattering for the defocusing masscritical nonlinear schrodinger equation for radial data in high dimensions tao, terence, visan, monica, and zhang, xiaoyi, duke mathematical journal, 2007. On uniqueness and stability for supercritical nonlinear. Aug 23, 2012 in these lecture notes we discuss the solution theory of geometric wave equations as they arise in lorentzian geometry. Nonlinear wave equations as limits of convex minimization. Sogge, lectures on nonlinear wave equations, monographs in analysis ii international press, 1995. Geometric and fourier methods in nonlinear wave equations. Courant lecture notes in mathematics, add to metacart. Jalal shatah, regularity results for semilinear and geometric wave.
The sharp estimate of the lifespan for semilinear wave equation with timedependent damping ikeda, masahiro and inui, takahisa, differential and integral equations, 2019 global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain sobajima, motohiro, differential and integral equations, 2019. The method of energy channels for nonlinear wave equations. Wave maps, i daniel tataru university of california, berkeley this is joint work with jacob sterbenz d. We consider the energycritical nonlinear focusing wave equation in dimension n 3, 4, 5. Scattering below critical energy for the radial 4d yangmills equation and for the 2d corotational wave map system. Ams proceedings of the american mathematical society. In particular, the interaction between the rapid normal. In the present article we study the dynamics at the critical level eu 0, u 1 ew, 0 and classify the. In a recent paper, struwe considered the cauchy problem for a class of nonlinear wave and schrodinger equations. Stochastic geometric wave equations 1939 on the latter, we refer the reader to nice surveys on geometric wave equations by shatah and struwe 57andtataru58. An explicit stationary solution, w, of this equation is known.
On uniqueness for supercritical nonlinear wave and. Stochastic geometric wave equations with values in compact. In this paper we consider the cauchy problem of the daveystewartson system on a circle. In a recent paper 39, struwe considered the cauchy problem for a class of nonlinear wave and schrodinger equations. Stochastic geometric wave equations 1939 on the latter, we refer the reader to nice surveys on geometric wave equations by shatah and struwe 57 and tataru 58. We establish, from a mathematical aspect, certain apriori estimates necessary to ensure the existence and uniqueness of solutions of the daveystewartson system model. Global infinite energy solutions of the critical semilinear wave equation. Geometric wave equations share this page jalal shatah. Library of congress cataloginginpublication data shatah, jalal m. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Geometric wave equations courant lecture notes by jalal shatah and michael struwe and a great selection of related books, art and collectibles available now at.
This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. A note on geometric heat flows in critical dimensions, submitted calculus of variations, 2006. Geometric wave equations, courant lecture notes 2, new york university 1998, 2nd edition. Geometric wave equations by jalal shatah michael struwe. Gallot hulin and lafontaine, riemannian geometry oneil, semiriemannian geometry. Exact solutions of semilinear radial wave equations in n. Geometric wave equations courant lecture notes by jalal shatah and michael struwe 20001024 jalal shatah and michael struwe on. The focus is on the work of the authors on semilinear wave equations with critical sobolev exponents and on wave maps in two space dimensions. Ihsan, 1957 geometric wave equations jalal shatah, michael struwe. Courant lecture notes in mathematics, new york university, courant. Schoen, a report on some recent progress on nonlinear problems in geometry, surveys in differential geometry 1 1991, 201241. Shatah wellposedness in the energy space for semilinear wave equations with critical growth, imrn 1994, no. Jalal shatah and michael struwe, geometric wave equations.
Struwe, geometric wave equations, courant lecture notes in mathematics 2 1998. The wave equation conservation laws function spaces the linear wave equation wellposedness semileaner wave equations wave maps wave maps with symmetry. Quintic wave equation 3 key properties of the shatah struwe solutions including the local and global existence, uniqueness and further regularity are collected in section 3. Wave maps into all hyperbolic spaces hn, which has been a longstanding and important conjecture in geometric wave equations due to its relation with problems in general relativity see chapter 16 of mr2473363 11. Selberg, multilinear spacetime estimates and applications to local existence theory for nonlinear wave equations, ph. Stochastic geometric wave equations with values in compact riemannian homogeneous spaces. Existence of classical solutions for a class nonlinear wave. Under some assumptions on the nonlinearities, it was shown that uniqueness of classical solutions can be obtained in the much larger class of distribution solutions satisfying the energy inequality. Solutions for a class nonlinear wave equations svetlin georgiev georgiev. Abstract ttin this article we investigate the cauchy problem for the equation. May 28, 2011 these notes are an extended exposion of lectures given at the conference nonlinear analysis, verbania, sept. We introduce spaces of finite energy sections and show wellposedness of the cauchy problem in those. Stable soliton resolution for equivariant wave maps exterior to a ball.
Christodouloutahvildarzadeh, shatah tahvildarzadeh, muller stru we, struwe, cote, cotekenigl. Geometric wave equations courant lecture notes in mathematics 1st edition by michael struwe, jalal shatah, jalal m. The proofs sketched in this exposition for global solvability of critical semilinear equations are essentially those presented in 20, 21. Regularity results for nonlinear wave equations annals. With the developement of general relativity and spacetime geometry, wave map equations, which defined on a curved lorentz manifold, have become more and more attractive. Geometric wave equations american mathematical society. American mathematical society, providence, ri, 1998. Another problem that we study in the manuscript is the cauchy problem for wave maps. Jalal shatah and michael struwe, geometric wave equations, courant lecture notes in mathematics, vol. The focus is on the work of the authors on semilinear wave equations with critical sobolev exponents and on wave. After receiving his doctorate in 1980, he was a member of the scientific staff in the special research sector 72 of the german research foundation and later an assistant at the mathematical institute of the university of bonn. When the data is not well prepared, we prove that the solution of the penalized equation converges weakly to the solution of the system of coupled equations obtained in 11 by a multiscale formal analysis.
Search for library items search for lists search for contacts search for a library. The focus is on the recent work of the authors on semilinear wave equations with critical sobolev exponents and on wave maps in two space dimensions. Pages 503518 from volume 8 1993, issue 3 by jalal shatah, michael struwe. A supercritical nonlinear wave equation in 2 space. The nonlinear wave equation 2 geometric nonlinear equations 3 wave maps 4 large data wave maps d. Singular perturbations of geometric variational problems, in. Geometric wave equations courant lecture notes by jalal shatah and michael struwe 20001024. Background material and references have been added to make the notes selfcontained. Struwe geometric wave equations courant lecture notes. Struwe, geometric wave equations, courant lecture notes in mathematics, 2. Scattering below critical energy for the radial 4d yang. Section 4 is devoted to the relatively simple subcritical case when the nonlinearity f grows slower than a quintic polynomial and the analogue of theorem 1. Thierry cazenave, semilinear schr odinger equations. New york university, courant institute of mathematical sciences, new york, ny, usa.
739 717 1585 1100 39 1141 812 300 1160 138 235 1047 1401 1613 742 251 949 1630 1657 1514 1081 1085 437 792 136 267 1392 1266 1433 384