Well-posedness result for the Kuramoto–Velarde equation

Abstract

The Kuramoto–Velarde equation describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface in a microgravity environment. Under appropriate assumption on the initial data, of the time T, and the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

On the Classical Solutions for the Kuramoto-Sivashinsky Equation with Ehrilch-Schwoebel Effects

“…it follows from ( 68), (69), ( 70) and (71) that (59), there exists ξ 4 between u 1 and u 2 , such that…”

“…In [56], the existence of solitonic solutions for (10) is proven. In [57][58][59][60], the well-posedness of the Cauchy problem for (10) is proven, using the energy space technique, the fixed point method, a priori estimates together with an application of the Cauchy-Kovalevskaya and a priori estimates together with an application of the Aubin-Lions Lemma, respectively (see also [61]). Instead, in [63][64], the initial-boundary value problem for (10) is studied, using a priori estimates together with an application of the Cauchy-Kovalevskaya, and the energy space technique, respectively.…”

About classical solutions for high order conserved Kuramoto-Sivashinsky type equation

“…In [50], the existence of solitonic solutions for (7) is proven. In [4,55,17,16,19], the well-posedness of the Cauchy problem for (7) is proven, using the energy space technique, the fixed point method, a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem and a priori estimates together with an application of the Aubin-Lions Lemma, respectively (see also [15]). Instead, in [14,37,38], the initial-boundary value problem for (7) is studied, using a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem, and the energy space technique, respectively.…”