Overview

Research Topic

Grant-in-Aid for Scientific Research(S)
Project Number : 25220702
Title of Project : Elucidations on uninvestigated problems related to the criticality of nonlinear dissipative and dispersive structures in mathematical models
Principal Investigator : Takayoshi Ogawa
JSPS Page

Purpose of our research

There are mathematical models in various fields of research (physics, biochemistry, astrophysics, electronics, plasma physics, etc.) that are formulated as coupled nonlinear partial differential equations. Our purpose is to investigate, via exhaustive analytical approach, the nonlinear phenomenon related to critical situations in regards to their dissipative and dispersive properties, and explore the possibility of nonlinear mathematics that is behind the critical structure.

Although we mainly focus on the study of nonlinear system of partial differential equations, we are also treating the related mathematical modelling problems, some inequalities or equations arising from differential geometry, geometric variational problems and nonlinear difference equations. In particular, our goal is to unlock the reason behind the models for natural phenomenon exhibiting certain criticalities and their tendencies to stabilize when the models reach their critical situations.

Background of research

As a typical example, it was considered difficult to solve integrable system such as KdV equation by means of analysis for very general data.

However, in 10 years following 1993, researchers such as Bourgain and Ponce established that the KdV equation is uniquely and stably solvable for a large class of of distributions including dirac delta measure. This type of problem is refered to as a ``problem of well-posedness'', and it is notable that the researchers in real analysis were great drivers in solving the well-posedness issue of the KdV equation (J. Bourgain and T. Tao are Fields medalists). As a byproduct of this theory, similar well-posedness results have been be established for problems that are not necessarily integrable systems.

On the other hand, in an argument for a collapse of 3-dimensional manifold via Ricci curvature flow which was used to solve Poincare conjecture, techniques derived from nonlinear partial differential equations played a significant role. In that scenario of solving Poincare conjecture, ``epsilon-regularity theory'' was utilized, which is important in the regularity theory for the incompressible Navier-Stokes system. If we apply wavelet analysis to such argument, there is a possibility of constructing a regularity theory for a large class of partial differential equations.

Relations with domestic researches

Taking in effective methods from real analysis, we may invoke some accumulation of knowledge. Indeed, some results concerning the improved regularity criterion for the incompressible Naveir-Stokes equations and rotating obstacle problems were obtained via Lettlewood-Paley type techniques. It seems plausible that some of classical researches were successful precisely because it founded upon real analytical method. Our past works indicate that researches involving the wave resonance in complex plane will become usefull in regards analyticity of a solution of some partial differential equations.

Purpose and expected influence of our research

In our research, our purpose is to deepen the understanding on mathemtical models expressed as nonlinear partial differential equations through rigorous techniques derived from real and complex analysis.

As described above, although deepening of understanding and its accessability may always be antithetical, often is the case where an easy solution to the dilemma is a simple and concrete example that effectively exhibits the significance of the problem. In mathematical physics or biochemistry, there are numerous such examples that help our understanding of nonlinear phenomenon in relation to purely mathematical knowledge.

For example, during Dictyostelium sporulation, a self-organization structure as well as a sphere formulation structure are observed and it is known that density concentration phenomena is related to a quantity appearing in the best constant 8π of isoperimetric inequality.

Dictyostelium sporulation

A discovery of a general theory that is applicable to not only differential equations but actual science has a tremendous impact upon many reaseachers including real analysts, probability theorist or differential geometricians.

In last 6 to 7 years, we have been putting efforts to discover the common properties of critical nonlinear PDE. Our experience tells us that real analytic theory founded on Littlewood-Paley theory is greatly effective for the analysis of such problems. The role of Littlewood-Paley theory to the nonlinear PDEs is as if that of psuedo-differential operators to the linear PDEs.

In actuality, techniques derived only from the pseudo differential operator theory cannot be usful for the analysis of critical type problems.

Therefore, obtaining via deeper understanding of real analysis a correspondence of pseudo differential operators that is applicable to general nonlinear theory is necessary for our study.

On the other hand, at the background of our problem, we have discovered that the criticality derived from the balance of dissipation/dispersion property and nonlinearity acts singularly.

In general, at a critical situation dissipation or dispersion stabilizes the system concerned, and this property seems to hold for any types of nonlinear system. As criticality is a product of nonlinear effect and linear stabilizing effect, and it is precisely because the problem is critical, we require some analytically end-point estiamtes. In order to obtain such estimates, it is important to grasp the global-in-time structure of solutions of the problem concerned. On such occation, what is significant is to clarify a structure of energy or entropy that are naturally defined from the model or to understand the (infinite-diemensional) geometric structure of variational functionals.

This line of study has been prominent for nonlinear heat equations for which the global behavior of solutions are directly connected to geometry of variational functional under prociple of least action. However, for physical models that are Hamiltonian and possess conservations laws, the variational formulation becomes contrained and the global behaviors of solutions greatly differ from those of dissipative equations.

It is an indication of our limited understanding of global structure of nonlinear problems that we obtain similar critical stabilization for different types of problems possessing different variational structures.

Understanding the variational structure of such models is one of important objectives in our research. As we said, it can be thought that in critical problems, nonlinear effect and dissipative/dispersive effect tend to stabilize the system complementarily.

We aim to clarify this phenomenon and present better understanding of mathematical structure of critical problems and it should become a starting point of developing new fields of analysis.

Comprehensiveness of our research and contributions from other related fields

In order to carry out our reasearch, we expect contributions from wide range of fields. Real analysis, differential geometry and probabilistic analysis are most probable to give birth to research that is not limited their fields. In particular, our research may deepen the understanding differential equations via probabilitic techniques related to branching process as well as real analysis. For example, there are many open problems that may be solved by an effective combination of PDE theory and probabilistic analysis of nonlinear dispersive equations via a noise perturbation. We focus on such open problem to instigate a new, exciting viewpoint.

Significance and speciality of our research

We point two factors that we especially attach importance.

The first factor is the discovery of critical functional inequality that are important in PDE theory. It is not an overstament to claim that everytime there is a discovery of a analytically precise inquality, more important products for PDE theory are born.

Here, among many general inequalities, we will primarily focus on investigation of certain critical Sobolev inequality, especially Sobolev type inequalities that arise from real interpolation theory to control the logarithmic function such as Gross' inequality and Brezis-Gallouet inequality as well as their duals which are Trudinger-Moser type inequalities. As natural development, we will examine the dissipative/dispersive estimates for linear PDEs with constant coefficients (so called Lp-Lq type estimates or Strichartz-Brenner type estimates). In addition to our previous work, this will require more precise bilinear estimates that are suited for nonlinear estimates.

Secondary, we investigate the global structure of nonlinear PDEs, in particular, the global geometric structure. We convert the problems with geometric background such null conditions for nonlinear hyperbolic equations by real analytical expression and describe it as certain analytical inequality. Development of analytical methods for problems with geometric background are extremely powerful for nonlinear problems which require very precise arguments. A noteworthy example of this fact would be divergence-curl lemma.

Purpose and background of our research

Many physical models can be expressed as nonlinear PDEs derived from interaction of physical quantities and are decomposed into a linear structure expressed by a partial differential operator and a nonlinear structure due to the influence of physical quantities. The linear structure consists of dissipative and dispersive structures and helps stabilizing the system and the nonlinear structure on the other hand makes the system unstable. We call a situation where the linear/stable structure and the nonlinear/unstable stucture become balanced a critical problem, which is our main tagert of research.

Both in mathematics and applications, many important problems possess a critical situation and present an interesting mathematical phenomenon. At critical situation, the perturbation method which is a main tool of analysis is not simply applicable, thereby making study more complicated.

Our principal aim of research is to investigate problems related to criticality and further establish a groundwork for the open region of super-critical problems.

Method of our research

We investigate in details in particular critical Sobolev inequalities, Gross' inqualities which control the logarithmic function, Brezis-Gallouet's inequalities and their dual counterpart Trudinger-Moser type inequalities. As natural development, we examine the dissipative/dispersive estimates for linear PDEs with constant coefficients (so called Lp-Lq type estimates or Strichartz-Brenner type estimates).

Expected products and significance

Many important problems are left open behind the critical problems, including some Millennium problems. In past works, there has been a case where one faces a situation in which a dissipative and dispersive effects annihilate each other and is not able to obtain a desired effect, thereby complicating mathematical proof. As a specially important tast, we expect that such difficulty will be solved as we uncover the dissipative and dispersive structure and the types of nonlinear effects. Moreover, establishing critical functional inequaties may lead to the polishing techniques of various analytical estimates interrelated to dissipative and dispersive effects such as bilinear/trilinear estimates, linear dissipative/dispersive estimates, maximal regularity principle.