A WEAK MAXIMUM PRINCIPLE AND A WEAK LIMIT FOR THE SOLUTION OF A FOURTH ORDER BURGERS EQUATION - ANALYTICAL RESULTS

摘要

In order to understand the effects of high order viscous regularizing terms on the solution of nonlinear PDEs, we consider the fourth order Burgers' equation u(t)(v)=u(v)u(x)(v) - vu(xxxx)(v) with 1-periodic initial and boundary conditions. We derive a maximum norm bound for the solution of the fourth order Burgers' equation, independent of the viscosity coefficient v, by obtaining bounds for the L(2i) norm of the solution of the equation, 1 less than or equal to i < infinity. We also obtain bounds for higher order derivatives of the solution. Using an iteration, we prove short-term existence; long-term existence is obtained using the short-term existence result and the a priori maximum bound. Finally, we prove that for t greater than or equal to 0, the fourth order Burgers' equation has a weak solution u in L(infinity)([0, 1]) and that the first order spatial and temporal derivatives of the weak limit u are measurable functions. [References: 6]