One of the most complex equations known in mathematics is the Navier-Stokes equation, which describes the motion of fluid substances such as liquids and gases. This equation is fundamental in fluid dynamics and is known for its complexity and the challenges it presents in finding solutions.
The Navier-Stokes equations are a set of nonlinear partial differential equations that can be expressed in the following form:
    ∂u/∂t + (u ⋅ ∇)u = -∇p + ν∇²u + f
Where:
u represents the velocity field of the fluid,
t is time,
p is the pressure field,
ν is the kinematic viscosity of the fluid,
f is the body force per unit mass (like gravity),
∇ is the gradient operator, and
∇² is the Laplacian operator.
The Navier-Stokes equations are notoriously difficult to solve, especially in three dimensions. In fact, one of the seven Millennium Prize Problems is to prove whether smooth solutions always exist for the three-dimensional incompressible Navier-Stokes equations, or if singularities can occur. A singularity would imply that the equations break down, leading to infinite velocity or pressure at some point in space and time.
Another example of a complex equation comes from general relativity, specifically the Einstein field equations, which describe how matter and energy in the universe influence the curvature of spacetime:
    Gμν + Λgμν = (8πG/c⁴)Tμν
Where:
Gμν is the Einstein tensor, which describes the curvature of spacetime,
Λ is the cosmological constant,
gμν is the metric tensor, describing the geometry of spacetime,
G is the gravitational constant,
c is the speed of light, and
Tμν is the stress-energy tensor, representing matter and energy content.
The Einstein field equations are a set of ten interrelated differential equations, and their solutions describe the gravitational effects of various mass distributions. They are fundamental to our understanding of cosmology and black holes, yet they can yield extremely complex solutions that are difficult to interpret.
In conclusion, both the Navier-Stokes equations and the Einstein field equations represent some of the most complex equations in mathematics and physics, challenging mathematicians and physicists alike. The ongoing research into these equations not only advances theoretical knowledge but also has practical implications in fields ranging from meteorology to astrophysics.
For further reading, consider these references:
Navier-Stokes Equations - Wikipedia
Einstein Field Equations - Wikipedia