In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, Strichartz estimates are a family of inequalities for linear
dispersive partial differential equation In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities.
Exa ...
s. These inequalities establish size and decay of solutions in
mixed norm
Mixed is the past tense of ''mix''.
Mixed may refer to:
* Mixed (United Kingdom ethnicity category), an ethnicity category that has been used by the United Kingdom's Office for National Statistics since the 1991 Census
* ''Mixed'' (album), a c ...
Lebesgue spaces. They were first noted by
Robert Strichartz
Robert "Bob" Stephen Strichartz (October 14, 1943 – December 19, 2021) was an American mathematician who specialized in mathematical analysis.
He was born in New York City on October 14, 1943. Bob graduated from Bronx High School of Science i ...
and arose out of connections to the Fourier restriction problem.
Examples
Consider the linear
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
in
with ''h'' = ''m'' = 1. Then the solution for initial data
is given by
. Let ''q'' and ''r'' be real numbers satisfying
;
; and
.
In this case the homogeneous Strichartz estimates take the form:
:
Further suppose that
satisfy the same restrictions as
and
are their dual exponents, then the dual homogeneous Strichartz estimates take the form:
:
The inhomogeneous Strichartz estimates are:
:
References
Theorems in analysis
Inequalities
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