Local well-posedness and regularity properties for an initial-boundary value problem associated to the fifth order Korteweg-de Vries equation

In this work we prove that the initial-boundary value problem (IBVP) for the fifth order Korteweg-de Vries equation {align*} . {array}{rlr} u_t+∂_x^5 u+u∂_x u&{-2mm}=0,& x∈ R^+,\; t∈ R^+,\\ u(x,0)&{-2mm}=g(x),&\\ u(0,t)=h_1(t),\, ∂_x u(0,t)&{-2mm}=h_2(t),\,∂_x^2 u(0,t)=h_3(t), {array} \} {align*} is locally well posed, when the data g, h₁, h₂, h₃ are taken in such a way that g∈ Hˢ( Rₓ⁺), and hⱼ₊₁∈ H^s+2-j5( Rₜ⁺), $j=0,1,2$, s∈ [0,114) \12,32,52\, and satisfy the following compatibility conditions: {align*} g(0)=h_1(0) if 12<s<32;\\ g(0)=h_1(0),\; g'(0)=h_2(0) if 32<s<52;\\ g(0)=h_1(0), \; g'(0)=h_2(0),\; g''(0)=h_3(0) if 52<s<{11}4. {align*} Besides, we prove that the nonlinear part of the solution is smoother than the initial datum g.