Application: A Nonlocal Burgers Equation

The equation we will investigate here is of the form (16.1)$$\frac{{du}}{{dt}} + N\left( n \right) = 0,{\text{ }}N\left( u \right) = Au + R\left( u \right),$$ on H={u∈2(0,L): ∫ ₀Lu(x)dx=0,0≤ x ≤L}, where (16.1a)$$R\left( u \right) = B\left( {u,u} \right) + \int , {\text{ }}\varphi {\text{ = 0,}}\psi {\text{ = }}f \ne 0,B\left( {u,\upsilon } \right) = \left( {u,\omega } \right)\upsilon ',{\text{ where }}\upsilon ' = d\upsilon /dx,{\text{ with a fixed }}\omega \in \ne {\text{0,}}$$$$A = - \frac{{{d^2}}}{{d{x^2}}}{\text{ with periodic boundary conditions}}{\text{.}}$$