My research interests (and possible PhD projects) fall into the following categories:

 

1. Partial differential equations (PDEs)

 

I am interested in various questions concerning the theory of linear and

nonlinear evolution partial differential equations. These include

 

Smoothing estimates for evolution PDEs

 

Many evolution PDEs have so-called smoothing properties. It roughly means that,

when averaged with respect to time, their solutions become more regular. This topic has

many very interesting relations to the harmonic analysis and spectral theory, as well

as to the well-posedness questions for nonlinear PDEs. Examples of equations are

hyperbolic equations (such as the wave equation, Klein-Gordon equation, dissipative

wave equations, as well as coupled equations and systems), Schrodinger, KdV, and

many others.

 

My research in this direction is mostly joint with M. Sugimoto, and the results in this

direction include the critical case of the Agmon-Hormander limiting absorption principle,

the critical cases of Ben-Artzi-Klainerman and Kato-Yajima smoothing estimates for the

Schrodinger and other equations, and estimates for norms for the critical cases of the

trace theorems. As an application, new global in time well-posedness results for the

derivative nonlinear Schrodinger equations have been obtained.

 

Moreover, extending the works by Ben-Artzi and Devinatz, Constantin and Saut,

Kenig, Ponce and Vega, and many other authors, in a series of papers with Sugimoto

we established a number of new smoothing estimates, and, more importantly, by

introducing new methods of canonical transforms and comparison principles for

evolution equations, we showed that essentially all known smoothing estimates for a

variety of a-priori unrelated evolution equations, are equivalent to each other and can

be obtained one from another (for example, the gain of half derivative for the Schrodinger

equation can be obtained from the gain of one derivative for the KdV equation, etc.).

My current work in this direction includes the extension of ideas of canonical transforms

and comparison principles to evolution equations with potentials and to equations with

variable coefficients (in particular, the interesting case is equations with potentials,

also allowing resonances at zero, which is a joint research with Matania Ben-Artzi and

Mitsuru Sugimoto).

 

Regularity, dispersive and Strichartz estimates for hyperbolic PDEs

 

Again, these are one of the main tools for the analysis of nonlinear PDEs. The analysis

here relies on the asymptotic properties and decay rates of solutions to linear equations.

The methods involve various estimates for so-called Fourier integral operators, which

is a class of operators of great interest by themselves. Especially interesting and challenging

are higher order equations or systems of equations, with multiple characterisitics.

I am also interested in applications of such analysis (e.g. to Kirchhoff equations),

as well as in their nonlinear perturbations and scattering theory.

 

A comprehensive analysis of dispersive estimates for hyperbolic equations of higher

orders with constant coefficients and for hyperbolic systems is carried out in the memoir

 

Ruzhansky M., Smith J., Dispersive and Strichartz estimates for hyperbolic equations with

constant coefficients, MSJ Memoirs, 22, Mathematical Society of Japan, Tokyo, 2010. (157pp)

 

The main difficulties here is the appearance of multiple characteristics of general form

and the results include the time decay estimates for different frequency zones (large frequencies,

small frequencies, frequencies around multiplicities, etc.) A certain geometric index is

associated to each equation, and the dispersive and Strichartz estimates are obtained

depending on this index, with the corresponding global in time well-posedness results

for hyperbolic equations. The results are applied to Grad systems of gas dynamics

(which are hyperbolic systems of large size depending on the number of moments),

and to the infinite dimensional systems appearing in the analysis of the Fokker-Planck

equations. In a series of papers with T. Matsuyama and J. Wirth the results are

extended to equations and systems with time-dependent coefficients, with the main

application to the Kirchhoff equations. The current research deals with equations with

variable coefficients and the corresponding geometric analysis. In particular, in this

direction can be mentioned the joint work with I. Kamotski where Sobolev space

estimates were established for a general class of hyperbolic systems with variable

multiplicities and the Weyl spectral asymptotics for the corresponding elliptic systems

(also with variable multiplicities) have been obtained.

 

2. Microlocal and harmonic analysis

 

Microlocal analysis can be viewed as an extension of the Fourier analysis for the study

of partial differential operators. It allows to describe various properties of operators and

functions, such as the propagation and location of singularities, questions of regularity,

smoothness, etc. I am mainly interested in questions related to the so-called

Fourier integral operators which is a fascinating class of operators that has relations to

many other areas of mathematics such as symplectic geometry, harmonic analysis,

spectral theory, PDEs, etc. Harmonic analysis aspects of interest to me are their

properties in Sobolev, modulation, Wiener amalgam, and other function spaces.

 

My results in this direction include the geometric analysis of the smooth factorization

condition for the Lp boundedness of Fourier integral operators introduces by Seeger, Sogge,

and Stein. Their Lp results for Fourier integral operators with real-valued phases as well as

the L2 results for Fourier integral operators with complex-valued phases of Melin and

Sjostrand, and Hormander, have been extended to the Lp estimates for Fourier integral operators

with complex-valued phases in the monograph

 

Ruzhansky, M. Regularity theory of Fourier integral operators with complex phases and

singularities of affine fibrations, CWI Tracts , vol. 131, Amsterdam, 2001. (136 pp)

 

Recently I came back to this topic in the global setting (partly motivated by the method of

the canonical transforms for the smoothing estimates), and the results include the global

(and weighted) L2 and Sobolev spaces results on the boundedness of Fourier integral

operators and the global calculus for such operators under minimal decay assumptions

(jointly with M. Sugimoto), global Lp results and the determination of weights

for the global Lp estimates for Fourier integral operators (jointly with S. Coriasco), as

well as results on the Beurling-Helsons theorems for modulation and Wiener amalgam

spaces and their relation to Fourier integral operators (joint with M. Sugimoto, N. Tomita,

and J. Toft).

 

Another result in this direction, motivated by applications to dispersive estimates, in the

multi-dimensional van der Corput lemma for oscillatory integrals (allowing complex

phases and the dependence of parameters), obtaining the multi-dimensional decay rate for

degenerate stationary points, thus bridging the gap between the one-dimensional decay rate

in the classical van der Corput lemma and the multi-dimensional decay rate for non-degenerate

stationary poins provided by the stationary phase method.

 

3. Pseudo-differential operators on Lie groups and manifolds

 

I am interested in the theory of pseudo-differential operators on spaces with symmetries,

such as Lie groups, symmetric, and homogeneous spaces, as well as more general manifolds.

The simplest case of a compact commutative Lie group is a torus (even a circle) and the

Fourier analysis of operators there is already very interesting. Here a global theory of

operators on such manifolds is of particular importance since it allows to capture many

geometric and other underlying algebraic properties of the manifold (a Lie group), and

to relate them to various properties of operators.

 

Some of my recent work in this area can be found in the monograph

M. Ruzhansky, V. Turunen, Pseudo-differential operators and symmetries, Birkhauser, 2010. (724pp)

This work is also closely related to the area of Time-Frequency Analysis

 

The work in this direction are joint with V. Turunen, and the main result here is the development

of the global quantisation of operators on compact Lie groups and homogeneous spaces

yielding globally defined matrix-valued full symbols and their calculus. Many properties of

operators can be expressed in terms of these symbols, and the knowledge of the full symbols

allows one to formulate manifold counterparts of well-known results on Rn that require the knowledge

of the full symbols (such as the sharp Garding inequality, etc., which require the knowledge of

the full symbol, while the standard theory of pseudo-differential operators via localisations

yields essentially only the principal symbols with an invariant global interpretation). The obtained

characterisation of the standard Hormander classes of pseudo-differential operators on manifolds

also allowed us to construct liftings of pseudo-differential operators from the homogeneous space to

a Lie group acting on it, thus extending the corresponding constructions well-known for invariant

differential operators, to the full pseudo-differential setting. The corresponding matrix-valued

quantisation of operators on S3=~SU(2) has been also linked to the explicit representation theory

of SU(2) and several related physical quantities (such as quantum numbers, Pauli matrices, etc.)

 

For my papers in these and other topics see my list of publications

 

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