The Well - Posedness Of Novikov Equation For Higher Order Deformation

This paper mainly investigates well-posedness of the high-order Novikov equation in the following formAfter analyzing Novikov equation which is obtained by using the symmetry ap-proach in the classification of non-local partial differential equations, and con-sidering the fact that the potential m in physics may have different forms, we establish the model through generalizing the potential m of the equation. To study the Cauchy problem, assume u0∈H2,p,P≥4, whereThe limit of viscous approximations for the higher-order Novikov equation is used to establish the existence of global weak solution. Because of cubic non-linearities of Novikov equation, it is complicated to generalize some conclusions, so the key point of this paper is the estimates of the cubic nonlinear terms and the process of the limit approximate solutions. We bound the viscous terms with the energy identity, and then get the convergence of the corresponding terms. Thanks to the regularity of the initial data, we acquire the estimate of the cubic nonlinearities, the convolution structure is introduced to deal with the higher-order derivation easily. What’s more, through the careful analysis of the structure of the equation, we obtain (?)x2uε converges to (?)x2u (strongly) in L2(R) Thereby we directly recover the existence of global weak solution.When the initial data is in different spaces, combined with the derivation of the existence of weak solution, we have the spaces are invariant under the action of equation, for each T> 0.When considering the uniqueness with the help of L∞estimate of trans-port equation, assume that uq ∈ H3(R), and if there exists a map b(t) ∈ L1([0,T]), T> 0, satisfies then the uniqueness of global weak solution of higher-order Novikov equation is obtained.