Moreover, due to the non-trivial background solutions, the induced metric of the relativistic string involves linear perturbations with undetermined signs, and hence a key observation is needed to guarantee that the energies associated to the multipliers are positive up to lower order terms. It then helps to proceed a geometrically adapted and weighted energy argument for which robust estimates suffice. By choosing a gauge adapted to the plane wave solution, we deduce an equivalent Euler-Lagrangian equation for the perturbation whose quasilinear structure is reflected precisely in the induced geometry of the relativistic string. Under certain decay assumptions on the plane wave, we conclude that the perturbed system admits a globally smooth solution if the perturbation is non-small along the travelling direction of the plane wave, while the perturbation along the transversal direction is sufficiently small. This paper is concerned with the global stability of the plane wave solutions to the relativistic string equation with non-small perturbations. It is noted that the methods and results in the paper can be extended to general 2D quasilinear wave equations satisfying corresponding null conditions for short pulse initial data. To overcome the difficulties due to the slower time decay rate of the solutions to the 2D wave equation and the largeness of the solution, we introduce some new auxiliary energies. The main ingredients of our analysis consist of showing the positivity of the inverse foliation density near the outermost conic surface for all time and solving a global Goursat problem inside the outermost cone. This is related to the Majda's conjecture on the non-formation of shock waves of solutions from smooth initial data for multi-dimensional nonlinear symmetric systems which are totally linearly degenerate. In this paper, we establish the global existence of smooth solutions to the 2-dimensional (2D) compressible isentropic irrotational Euler equations for Chaplygin gases with the short pulse initial data introduced by Christodoulou. In the seminal work, Christodoulou introduced the "inverse foliation density" µ to measure the compression of the outgoing characteristic surfaces, and proved the finite time formation of shocks for 3D relativistic Euler equations with small initial data by developing a geometric approach which has been applied and refined to study shock formation for other important problems, see, where the key is to show that µ is positive away from the shock and approaches 0 + near the blowup curve in finite time based on the genuinely nonlinear conditions. Christodoulou in order to study the formation of shocks for multi-dimensional hyperbolic systems and the second order wave equations with the genuinely nonlinear conditions, see also. Our analysis is strongly motivated by the geometric approach initiated by D. Finally, we make some brief comments on the analysis of the proof of Theorem 1.1, the details are given in the next subsection. Quasilinear wave equations in three space dimensions, small-data shockįormation occurs precisely when the quadratic nonlinear terms fail the classic
Speck, whichĮxtends Christodoulou's result to show that for two important classes of Result, outline its main new ideas, and place it in the context of the above Our goal in this paper is to give an overview of his John in the midġ970's and continued by S. Higher-dimensional quasilinear wave equations, initiated by F. Large body of prior work concerning the long-time behavior of solutions to His work provided a significant advancement over a ($s$ sufficiently large), in solutions to the relativistic Euler equations in Christodoulou proved a remarkable result giving aĭetailed description of shock formation, for small $H^s$-initial conditions This is the transport equation.In his 2007 monograph, D. I want to solve the following Partial Differential Equation by using the method of characteristics.