Nonlinear wave equations and singular solutions [PDF]
Proceedings of the American Mathematical Society, 135 , no. 11,
3659-3667 (2007), 2006 We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed.
arxiv
Well-posedness for a modified Zakharov system [PDF]
arXiv, 2006 The Cauchy problem for a modified Zakharov system is proven to be locally well-posed for rough data in two and three space dimensions. In the three dimensional case the problem is globally well-posed for data with small energy. Under this assumption there also exists a global classical solution for sufficiently smooth data.
arxiv
Existence of Metrics with Prescribed Scalar Curvature on the Volume Element Preserving Deformation [PDF]
arXiv, 2006 In this paper,we obtain two results on closed Reimainnian manifold $M\times [0,T]$.When $T$ is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When $T$ is large and the given scalar curvature is small enough,the same result holds.
arxiv
Differential Invariants and Construction of Conditionally Invariant Equations [PDF]
arXiv, 2003 New concept of conditional differential invariant is discussed that would allow description of equations invariant with respect to an operator under a certain condition. Example of conditional invariants of the projective operator is presented.
arxiv
Blow-up results for some second order hyperbolic inequalities with a nonlinear term with respect to the velocity [PDF]
arXiv, 2005 We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some semilinear systems of mixed type.
arxiv
On reduction of multi-dimensional non-linear wave equation to two-dimensional equations [PDF]
Reports of the Acad. Sci. of Ukraine (Dokl. Akad. Nauk Ukr. SSR),
Ser. A, Phys., Math and Tech. Sciences, 1990, No. 8, p. 31--33, 2006 A condition of reduction of multidimensional wave equations to the two-dimensional equation is studied, and the necessary conditions of compatibility and exact solutions of the resulting d'Alembert-Hamilton system are obtained.
arxiv
Local Existence for Nonlinear Wave Equation with Radial Data in 2+1 Dimensions [PDF]
arXiv, 2007 We get a local existence result in $H^s$ with $s>3/2$ for second order quasilinear wave equation with radial initial data in 2+1 dimensions, based on an improvement of Strichartz estimate in the radial case. Moreover, we get the corresponding local well-posed result for semilinear wave equation.
arxiv
Low Regularity local well-posedness for the 1+3 dimensional Dirac-Klein-Gordon system [PDF]
arXiv, 2007 We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1+3 dimensions is locally well-posed in a range of Sobolev spaces for the Dirac spinor and the meson field. The result contains and extends the earlier known results for the same problem.
arxiv
Gradient NLW on curved background in 4+1 dimensions [PDF]
arXiv, 2008 We obtain a sharp local well-posedness result for the Gradient Nonlinear Wave Equation on a nonsmooth curved background. In the process we introduce variable coefficient versions of Bourgain's $X^{s,b}$ spaces, and use a trilinear multiscale wave packet decomposition in order to prove a key trilinear estimate.
arxiv
Smoothing estimates for the Schrodinger equation with unbounded potentials [PDF]
arXiv, 2008 We prove a local in time smoothing estimate for a magnetic Schrodinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two derivatives. The proof is based on the multiplier method and no pseudofferential techniques are required.
arxiv