Dorothy Moorefield

Women
and Minorities in Mathematics

Dr.
Sarah Greenwald

29
March 2001

OL'GA ALEKSANDROVNA LADYZHENSKAYA

Ol’ga Aleksandrova
Ladyzhenskaya was born on March 7, 1922. The patrons of her hometown, Kologriv,
must be proud to claim her as their own.[1]
She proves to be a lovely counter example to Hardy’s quote, “Mathematics is a
young man’s game.” Her contributions to the fields of Mathematics and Physics
are beyond the scope of a single paper.

O. A. Ladyzenskaya
received her undergraduate degree from Moscow State University in 1947. In
between then and 1949 she furthered her career by earning her PhD from the
Leningrad State University. In
1943 she earned yet another degree from Moscow State University this being a
Doctorate of Sciences. Two years later she became a Professor of Mathematics at
the Physics Department of St. Petersburg University.[2]
She can be reached at the Laboratory of Mathematical Physics in the Steklov
Institute of Mathematics at St. Petersburg where she has been presiding as the
head of the Laboratory since 1961. [3]
[4]

O.A. Ladyzenskaya
coauthored the book, __Linear and Quasilinear Equations of Parabolic Type__,
with V. A Solonnikov and N.N. Uranl’ceva.
This book is based on linear, quasi-linear second-order partial
differential equations of parabolic type.[5]
A partial differential equation contains partial derivatives depending on more
than one variable. To solve a partial differential equation one seeks the
solution whose derivatives when plugged into the equation are satisfied. The
solution is most likely not a constant value but a function of the variables.
The order of a differential equation is given by the highest partial derivative
within the equation. For example, the order of the wave equation in three
dimensions represented by, U_{tt} =U_{xx} + U_{yy} + U_{zz},
is 2. This equation is also linear.[6]

A Linear Partial Differential equation is basically an
equation whose dependent variable, which is the solution to the equation, and
its derivatives are given in a linear form. Basically this means the unknown
dependent variable along and its derivatives, are given to be in a linear
combination within the equation. For example here is a second-order linear
partial differential equation with two independent variables, A*U_{xx}
+ B*U_{xy} + C*U_{yy }+ D*U_{x} + E*U_{y} + F*U
= G. A, B, C, D, E, F and G are either constants constant coefficients or they
are functions of x and y. In order for the equation to be linear, neither U nor
its derivatives taken to any power, multiplied together or plugged to a function
of any kind such as sine or cosine.[7]

Equations of the parabolic type are given by
the discriminate of the second-order linear partial differential equation. The
discriminate is just like the discriminate of the quadratic formula in
appearance, B^{2} – 4*A*C. If the coefficients of the partial
differential equation have the property, B^{2} – 4*A*C=0 then the
equation is parabolic. Another type of linear partial differential equations is
elliptic. If the discriminate is less than zero, then the equation is elliptic.
[8]
O.A. Ladyzenskaya wrote another book with N. N. Ural’ceva entitled, __Linear
and Quasi-Linear Equations of Elliptic Type.__[9]

Quasi-linear equations
are not to be confused with linear equations. In fact quasi-linear equations
are nonlinear. However, quasi-linear means almost or somewhat linear. There are
two main interpretations of what “almost linear” means. One interpretation is
the coefficients on U and or its derivatives are nonlinear but their value is
so small, the nonlinear part of the equation is almost insignificant. The other
interpretation is what O.A Ladyzhenskaya appears to follow so we shall focus on
equations on this type.[10]

The equations in discussion are a special
type of nonlinear equations which take the general form of L U º U_{t} –
a_{ij}(x , t, U, U_{x})*U_{xixj } +a( x, t, U, U_{x}) = 0.[11] L is the linear operator on U, which sets up
the equation. L U is defined to be U_{t} – a_{ij}(x
, t, U, U_{x})*U_{xixj } +a( x, t, U, U_{x}) = 0. The coefficients, a_{ij}(x
, t, U, U_{x}) and a( x, t, U, U_{x}) may be dependent on U or
its derivatives not with respect to time. However, the partial derivative with
respect to time, U_{t}, is a linear term. This provides the other
interpretation of what it means to be quasi-linear.[12]

To understand the notation of this equation
one must conceptualize many dimensions. The equation is set into Euclidean
space with n dimensions. The variable x stands for n variables where, x = (x_{1
}, …, x_{n}). In two dimensional space x = ( x_{1 }, x_{n})
which is commonly notated by (x , y). U_{x} stands for the first partial
derivatives of U with respect to all variables. So we have, U_{x} = ( U_{x1},
… ,U_{xn} ). Comparing once again to two dimensional space, we have U_{x}
= (U_{x1} , U_{x2}) which is commonly notated as ( U_{x}
, U_{y}). U_{xixj} notates the partial derivative with respect
to x_{i}, some arbitrary independent variable, and then the partial of
that derivative taken with respect to x_{j}, which is another arbitrary
independent variable. The variable t usually represents time and is not within
our n dimensions of Euclidian space. The terms, a( x, t, U, U_{x}) and
a_{ij}(x , t, U, U_{x}) are the coefficients which my be
functions dependent on one to all of the following terms; The term x, denotes a
point in n dimensional Euclidian space and can be read as x = (x_{1 },
…, x_{n}). Time, denoted by t, is not in our n dimensions. U_{t}
is the partial derivative of U with respect to t. U is our unknown dependent
variable. U_{x }represents the first derivative with respect to all
variables within our n dimensions where, U_{x} = ( U_{x1}, … ,U_{xn}
).[13]
If the coefficients are constants or dependent only on x and t, then the
equation is now linear.

Quasi-linear equations have properties
different than linear equations. The properties that make them hyperbolic,
parabolic or elliptic are quite different from the discriminate use to
determine these traits in linear equations. Due to time we shall skip these
properties and move on to some examples of quasi-linear equations.

Consider the Cauchy problem for systems of
quasi-linear equations. U_{t }= P(x, t, u, U_{x})*U + F(x,t), xÎR^{n}
, 0#t#T. xÎR^{n } shows that we are in n dimensions. P(x, t, u, U_{x})=j_{1}_{v}_{1}_{#}_{m} A_{v}(x, t, U)*^{,}^{v}^{,}/ (*x_{1}^{v1} … *x_{n}^{vn}).
The function, F, is the forcing function and is assumed to be known. U_{t} is a linear term and if
it is moved across the equality the equation looks quite similar to the general
form. This equation is accompanied by an initial the condition, U(x, 0)= f(x)
where xÎR^{n}
and the initial function, f(x), is assumed to be known.[14]

Another example is the inviscid Burgers’
equation: U_{t} + (1/2)*(U^{2})_{x} = 0. Once again the
equation is not completely linear but U_{t} is a linear term thus
making the equation quasi-linear. This is a special case of the Navier-Stokes
equations.[15]

Navier-Stokes
equations are used to model the behavior of fluids. O.A. Ladyzhenskaya’s work
on the Navier-Stokes equations has had a profound effect on the field of Fluid
Dynamics.[16] She has
inspired numerous papers and has received some recognition such as conferences
held in her honor[17]
and contributions dedicated to her.[18]
Her father encouraged her along with many of her colleages.[19]
Their encouragement and inspiration were not fruitless. Future generations may
very well aknowledge Ol’ga Ladyzhenskaya as one of the most influential people
in her field.

** **

Bibliography:

1) http://www.awm-math.org/noetherbrochure/Ladyzhenskaya94.html

2) http://www.pdmi.ras.ru/staff/ladyzhenskaya.html

3) Ladyzhenskaya, O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. __Linear
and Quailinear
Equations of Parabolic Type.__
Translated by Smith, S.. American Mathematical Society. Providence,
Rhode Island. 1968.

4) Farlow, Stanley J.. __Partial Differential Equations for
Scientists and Engineers.__ Dover Publications, Inc.. New York. 1982.

5) Discussion with Dr. Eric S. Marland, Assistant Professor of Mathematical Sciences at Appalachian State University.

6) Kreiss, Heinz-Otto; Lorenz, Jens. __Initial-Boundary
Value Problems and the Navier-Stokes Equations.__ Academic press INC..
Boston. 1989.

7) http://pore.csc.fi/math_topics/Mail/NANET98-2/msg00033.html

8) http://www.mat.uni.torun.pl/~tmna/htmls/archives/vol-9-1.html

9) http://www.mat.uni.torun.pl/~tmna/htmls/archives/vol-9-1.html

10) Ladyzhenskaya, O.A.. __The Boundary Value Problems of
Mathematical Physics.__ Translated by Jack Lohwater. Springer-Verlag, New
York. 1985.

[5]
Ladyzhenskaya, O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. __Linear and
Quailinear Equations of Parabolic Type.__ Translated by Smith, S.. American Mathematical Society.
Providence, Rhode Island. 1968

[6] Farlow,
Stanley J.. __Partial Differential Equations for Scientists and Engineers.__
Dover Publications, Inc.. New York. 1982.

[7] Ibid.

[8] Ibid.

[9]
Ladyzhenskaya, O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. __Linear and Quailinear
Equations of Parabolic Type.__
Translated by Smith, S.. American Mathematical Society. Providence,
Rhode Island. 1968

[10] Discussion with Dr. Eric S. Marland, Assistant Professor of Mathematical Sciences at Appalachian State University.

[11] Ladyzhenskaya,
O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. __Linear and Quailinear Equations of
Parabolic Type.__ Translated by
Smith, S.. American Mathematical Society. Providence, Rhode Island. 1968

[12] Discussion with Dr. Eric S. Marland, Assistant Professor of Mathematical Sciences at Appalachian State University.

[13] Ibid.

[14] Kreiss,
Heinz-Otto; Lorenz, Jens. __Initial-Boundary Value Problems and the
Navier-Stokes Equations.__ Academic press INC.. Boston. 1989.

[15] Ibid.

[17] http://pore.csc.fi/math_topics/Mail/NANET98-2/msg00033.html

http://www.mat.uni.torun.pl/~tmna/htmls/archives/vol-9-1.html

[19]
Ladyzhenskaya, O.A.. __The Boundary Value Problems of Mathematical Physics.__
Translated by Jack Lohwater. Springer-Verlag, New York. 1985.

http://www.awm-math.org/noetherbrochure/Ladyzhenskaya94.html