arXiv research

The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach they obtained the l…

Study proves global existence and decay for complex wave equations.

Global existence and boundedness proved for quasilinear wave equations on Kerr black holes.

Global existence and decay for quasilinear wave equations on various spacetimes, including Kerr black holes.

In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is int…

Stability of catenoid in hyperbolic space proven without symmetry assumptions.

This is the first in a series Of papers in which we initiate the study Of very rough solutions to the initial value problem for the Einstein Vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques Of energy estimates and Sobolev in…

Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed. These results were recently applied by I.M.Krichever and B.A.Dubrov…

Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.

Study critical quasilinear equations on Riemannian manifolds with curvature constraints.

A quadratic line complex is a three-parameter family of lines in projective space P^3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P^3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P^3 with a c…

Sharp estimates derived for quasilinear equations on metric measure spaces.

Stability of catenoid in 4D Minkowski space proven without symmetry assumptions.

Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.

Proves regularity for quasilinear elliptic equations in metric spaces.

Based on Lie group method, potential symmetry and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in explicit form, we focus on the physically interesting situations which admit potential symmetries. Then by using the partial Lagrangian approach, we fi…

The paper proves growth estimates for subsolutions of quasilinear equations.

Study eigenvalue problems on Riemannian manifolds with specific curvature conditions.

Estimates for solutions on manifolds under Ricci flow.

We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with $n$ independent and $m$ dependent variables ($n\times m$ systems). We solve the symmetry conditions in a geometric way and determine the general form of the symmetry vector and of the Noetherian conservation …

We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, …

The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold $M$ with boundary $\partial M$. We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on $M$. We propose …

We show, by modifying Borbély's example, that there are $3$-dimen\-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $\le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.

Paper analyzes solutions to quasilinear elliptic equations on manifolds using Nash-Moser iteration.

For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.

The paper classifies solutions to a specific elliptic equation in the Heisenberg group.

Study curve shortening flow on Riemann surfaces with conical singularities.

Paper explores non-uniqueness and uniqueness class for wave equations on graphs.

New method finds precise late-time behavior of wave equations.

We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of linearizability of characteristic webs on all solutions imposes simple second-order con…

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets o…

Classifies solutions to vacuum weighted Einstein equations on pr-waves.

Unique solutions found for wave-like decaying null infinity equations.

Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.

Geometric flow on curves in S^3 generates YO equations solutions.

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…

New method of symmetrization applied to PDEs on spheres.

The paper solves a complex financial optimization problem using a novel mathematical technique.

Study proves well-posedness and scattering for wave equations on hyperbolic spaces with singular data.

Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.

The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…

Proves wave equation solutions in Kerr-de Sitter spacetime have specific asymptotic expansions.

In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell's equations. These results are analogous to those obtained by the a…

Measuring wave sources uniquely identifies manifold properties.

Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.

We study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough.

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…

The paper establishes scattering theory for wave equations on Schwarzschild spacetime.