represent the velocity field as: where is the number of elements, and are the shape functions associated with each element. Substituting this expression into the Navier-Stokes equation and applying the finite element method, we can derive a system of ordinary differential equations: where is the domain, and the integrals are over the domain. This system of ordinary differential equations can be solved using techniques such as the finite element method or spectral methods. Such method can be applied as: we can use a variety of techniques, such as the finite element method or spectral methods. One common approach is to use a finite difference method, which involves approximating the derivative terms in the equation using finite differences. To do this, we can divide the time interval into a series of smaller time steps, and approximate the derivative at each time step using a finite difference formula: where is the size of the time step, and and are the values of and at time step . Using this approximation, we can iterate through the time steps and compute the value of at each time step. For example, starting at time step and using the approximation above, we can compute the value of at time step : This process can be repeated until we reach the final time step . There are many other approaches to solving ordinary differential equations, each with its own advantages and disadvantages. The choice of approach depends on the specific equation being solved, and the desired accuracy and efficiency of the solution. Two settings: unbounded and periodic space There are two different settings for the one-million-dollar-prize Navier–Stokes existence and smoothness problem. The original problem is in the whole space , which needs extra conditions on the growth behavior of the initial condition and the solutions. In order to rule out the problems at infinity, the Navier–Stokes equations can be set in a periodic framework, which implies that they are no longer working on the whole space but in the 3-dimensional torus . Each case will be treated separately. Statement of the problem in the whole space Hypotheses and growth conditions The initial condition is assumed to be a smooth and divergence-free function (see smooth function) such that, for every multi-index (see multi-index notation) and any , there exists a constant such that for all The external force is assumed to be a smooth function as well, and