## Research

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#### Toward a World-class Research Base with

Innovative Approaches to Mathematical Fluid Dynamics

Research Institute of Nonlinear Partial Differential Equations

Since olden days, people have tried to understand nature through mathematics. Scholars active in the 16th to 18th centuries, starting with Galileo Galilei who left us with the words of "The greatest books of nature have been written in the language of mathematics," as well as Kepler, Newton, Leibniz, and others, tried to explain each natural phenomenon by mathematical laws. It was Euler in the 18th century who first tried to describe nature through nonlinear partial differential equations. We can say that the development of modern-day civilized society and science and technology that has been driven by industrial society has had this basis supported through basic partial differential equations amassed by our predecessors.

Partial differential equations up to the 20th century achieved advanced development in the "linear" world. That is, analysis of the world by proportional or inversely proportional cause and effect. From the late 19th century, laws of continuous natural phenomenon from inside nature were variously broached and partial differential equations were applied as principles of engineering. Moreover, the development of industrial technology gave birth to new partial differential equations. Engineering is an academic subject which is not only about the application of physics and mathematics to technology; it is also about organizing real technology using mathematics. From the beginning, mathematics and engineering have developed in close relation. In particular, industrial technology that developed dramatically in the 20th century was built on an understanding of linearity.

After this, as we have met the 21st century, we are now in an age where we are undergoing a paradigm shift from linear to nonlinear. For example, we are witnessing the reality of the nonlinear like the economy which had been growing in a straight line (linear), but then underwent a complete change thrusting to the very bottom of the recession. We have now reached a stage where everyone recognizes the importance of understanding nonlinearity. In the linear, complicated phenomenon which seemingly cannot be solved were described as more comprehensive phenomenon including marginal fluctuations. So nonlinear partial differential equations have been explored and we have entered a phase of the development of new engineering and new industrial technologies which became possible through nonlinear partial differential equations.

Among these global trends, the Research Institute of Nonlinear Partial Differential Equations was established as one research base of the Organization for University Research Initiatives at Waseda University in October 2010, with the aim of becoming a base which will lead the cutting edge of this. Based on the challenge of understanding nonlinear partial differential equations and their engineering applications, researchers within the university have gathered beyond the barriers of faculty or academic discipline and have started the institute as a base of strategic collaborative research in order to open up the frontier of the 21st century. We talked to Professor Yoshihiro Shibata of the Faculty of Science and Engineering who works as the director of this institute about this initiative.

The flight motion of a butterfly which creates a vortex of air by the flapping of its wings showing complex motions is one nonlinear motion

From among the members of the institute, Director Yoshihiro Shibata, Professor Hiroaki Yoshimura, Professor Katsuhiro Yamamoto and Senior Researcher Yukihito Suzuki, all of the Faculty of Science and Engineering (from left in the photograph)

##### Targeting Fluidity with All-out Effort

Nonlinear mathematics makes it possible to scientifically describe more complex dynamic principles that are found in biological and social phenomenon. It is expected that through engineering, new mathematics will bring approaches different to those before in all science and technology and industrial fields; not only the manufacturing industrial fields of energy, chemistry, steel, resources, aerospace, public works and construction, but also everything from biology and medical care to the global environment and financial engineering. It is anticipated this will bring about a wide range of innovations from product development to social systems. Through collaborations between mathematics and engineering, the elucidation of partial differential equations which become the mathematical foundation and the advance of applications to the engineering field at the same time will become a prerequisite to go to the top in this area. Accordingly, while traversing the foundations and applications, the organization of cooperation, where the expertise is utilized together, will be the key to growth.

"The development of nonlinear partial differential equations and engineering where nonlinear partial differential equations have been applied is, without mistake, a frontier area of humanity in the 21st century. While global competition between research bases in countries all over the world has begun to unfold, we, the Waseda University team, are aiming to form a base that will be one of the best in the world." (Director Yoshihiro Shibata)

The strength of Waseda University is first this thickness in the layer of researchers. In the Research Institute of Nonlinear Partial Differential Equations ten people have gathered from each field of mathematics, applied physics, engineering and more from across the whole university. These are exceptional persons that take approaches from various angles in the entire nonlinear world; from basic to applied research.

"If there is a university which has special study of mathematics and engineering, then it is to be expected that there will always be some teachers researching the area of nonlinearity, but I think that there are no other universities in this country with as many and as diverse researchers in this field as this university. However, until now, these researchers were scattered here and there in different faculties and majors and their laboratories were also in different places on campus. This meant that we had few opportunities to form connections. Through the establishment of this institute, it has been possible to create a place where the members can actually get together and carry out research. With this opportunity, we will expand cooperative research at a stroke."

Organization of the Research Institute of Nonlinear Partial Differential Equations

Researchers in relevant fields who had been divided across the university have been brought together under one roof and have formed a strong system for collaborative research

Hokkaido University, Tohoku University, Tokyo University, Osaka University and also major national universities are pouring their efforts into research on nonlinear partial differential equations, and it can be said that this area is attracting a lot of attention. However, among these, this research institute of Waseda University is characterized as a place where there is not only mathematical research, but also close cooperation with engineering application areas.

"Our team has brought together many members with strengths in research that from the start has targeted fluid dynamics. Accordingly, among nonlinear phenomenon, it is especially fluid dynamics that we are targeting and we are looking to this by an all-out effort with different fields working in cooperation. Even though fluid dynamics is one phrase, there is a broad range of applications. Of course these fields include space, aviation, meteorology, marine and others, but in terms of health care there are also related areas, such as the flow of blood and the flow of everything. In the field of mathematical fluid dynamics, research is taking place that can only be done at this university. We are establishing a research base that will lead not just Japan, but the whole world."

Waseda University has strengths in areas relevant to fluid dynamics, the Schr旦dinger equation, nonlinear elliptic equation, mathematical fluid dynamics, engineering fluid dynamics and numerical simulation. In this institution, it is the important field of mathematical fluid dynamics that has been set as the target of research.

##### Taking on the Challenge of an Unresolved Problem from 150 Years Ago

Mathematical fluid dynamics has great potential to bring about innovation in the world of manufacturing. For example, in research where base metals and precious metals are made to chemically react, during the flow, the occurrence of a stable reaction has been understood experimentally. Moreover, mechanisms which generate bubbles and the behavior of the bubbles in the flow are also a topic that has been gaining attention.

While fluid motion is something that is familiar to us, such as the flow of air and water, predictions of this are extremely difficult. This is especially remarkable when compared to the fact we can accurately predict decades into the future solar and lunar eclipses in the distant universe. This is because there are many factors working together that have an effect on flow, for example variations due to differences in velocity (nonlinear advection), the effect of particles jostling amongst each other (pressure), the effect of high velocity fluid dragging along low velocity fluids and barriers (viscosity) and buoyancy caused by variations in density and temperature.

As the most basic nonlinear partial differential equation describing a fluid dynamic phenomenon like this, we have the Navier-Stokes equation which was proposed around 150 year ago. To this day, airplane and automobile wind tunnel simulations are conducted based on this equation. However, in fact, this Navier-Stokes equation is one that is still yet unresolved in mathematics.

This is the Navier-Stokes equation which was proposed 150 years ago. As a Millennium Problem, there is a one million dollar prize for solving this equation.

"The Navier-Stokes equation was selected as one of the 'seven mathematical problems that should be solved in the 21st century,' which was established in 2000. A one million dollar prize will be awarded for solving this problem. Taking on the challenge of this difficult question called the Millennium Problem is one of our goals." (Director Yoshihiro Shibata)

##### Leading the World in Bridging the Meso and Macro

A research meeting held to commemorate the establishment of the institute November 9, 10 and 11, 2010). Leading researchers from home and abroad were invited and mini-courses and lectures took place.

In order to research complex flow like that which is gaining attention in applied fields, it is becoming necessary combine research on a macro level (which describes motions such as can be seen by the human eye like the Navier-Stokes equation) and research on a micro level (like that which cannot even be seen when using a microscope). It is well-known that all things are made of atoms and molecules. We also understand that the motions of these are described by the Hamilton-Jacobi equation. However, when these move after taking the form of a fluid as an aggregation of a vast number (e.g. around 1024), how this will come to follow the Navier-Stokes equation is in fact not completely understood.

"Challenges come with directly combining micro level equations with macro level equations. Consequently, in order to tie together the macro and micro we have set several targets in the meso level which exists between this. For example, like the behavior of bubbles generated and extinguished in extremely intense water currents, we are focusing attention on phenomena which are not completely sufficiently grasped with just a macro perspective and we are searching for this mechanism with the collaborative research of mathematics and engineering." (Director Yoshihiro Shibata)

Mathematical fluid dynamics has great potential to bring about innovations in the world of manufacturing, which is an area Japan triumphs in. Recently, the generating mechanism of nano-bubbles which have been discovered to have various uses and their behavior in flow are topics that have been acquiring interest.

"We understand experimentally that if we add vibrations at a certain frequency, small bubbles in water will reach a stable state. However, the same has also been found in results where a model describing the phenomenon is mathematically analyzed. Furthermore, for example, it has been confirmed experimentally that if we better control the flow, we are able to efficiently and accurately perform precious metal plating on minute particles. It is one of our aims that if we mathematically understand these mysterious natural phenomena, to apply these features in engineering and utilize them in new manufacturing technologies."

Then, Director Yoshihiro Shibata brought his explanation to a close. "In this way, we have set ourselves original goals unprecedented in the world of a quest on the meso level that will become a bridge between the differing micro and macro levels, as well the establishment of a rigorous mathematical theory of the level structure for the micro and macro. We are tackling these targets with a strong spirit that leads the world of mathematical fluid dynamics.

Efforts are also being poured into international academic exchanges and information dissemination. Since 2009, Waseda University has been participating along with the Technische Universitat Darmstadt of Germany in the International Research Training Group (IGK 1529) (Internationales Graduiertenkolleg) Mathematical Fluid Dynamics and leading international cooperative education in this field is being developed. In research establishments, international conferences and research meetings are held by top-level researchers from around the world and personnel exchange is encouraged. Together with this, the group is pursuing a role as an educational base to the younger generation, for example by inviting leading researchers from overseas to come and give mini-courses. There are great expectations the institution will contribute to the development of new engineering as well as new science and technology backed by new mathematics in the 21st century.

###### Related Links

Faculty of Science and Engineering, Waseda University

The International Research Training Group (IGK 1529) (Internationales Graduiertenkolleg) Mathematical Fluid Dynamics (Waseda University - Technische Universitat Darmstadt)