On higher-order viscosity approximations of odd-order nonlinear PDEs

Abstract

Some aspects of vanishing viscosity (e → 0+) approximations of discontinues solutions of odd-order nonlinear PDEs are discussed. The first problem concerns entropy solutions of the classic first-order conservation law (Euler’s equation) (0.1) $$u_t + uu_x = 0 \quad({\rm or}\, u_t + u^2 u_x = 0),$$ which are approximated by solutions u e (x, t) of the higher-order parabolic equation (0.2) $$u_t + uu_x = \varepsilon (-1)^{m+1} D_x^{2m} u, \quad D_x = \partial/\partial{x}, \quad {\rm with\, integer}\, m \ge 2.$$ Unlike the classic case m = 1 (Burgers’ equation), which is the cornerstone of modern theory of entropy solutions, direct higher-order approximations of many known entropy conditions and inequalities are not possible. By use of the concept of proper solutions from extended semigroup theory, it is shown that (0.2) and other types of approximations via 2mth-order linear or quasilinear operators correctly describe the solutions of two basic Riemann problems for (0.1) with initial data $$S_\mp(x) = \mp{\rm sign}\, x,$$ corresponding to the shock (S −) and rarefaction (S +) waves, respectively. The second model is taken from nonlinear dispersion theory with the parabolic approximation (0.3) $$u_t - (u u_x)_{xx}= \varepsilon (-1)^{m+1} D_x^{2m} u, \quad {\rm with} \, m \ge 2.$$ Similar evolution properties of e-approximations of stationary shocks S ±(x) posed for (0.3) are established. Special “integrable” quasilinear odd-order PDEs are known to admit non-smooth compacton or peakon-type solutions (e.g., the Rosenau–Hyman and FFCM equations), while for more general non-integrable PDEs such results are unknown. It is shown that the shock S −(x) for (0.3) is obtained as e → 0 by an ODE approximation and also via blow-up self-similar solutions focusing as t → T −. For S +(x), the corresponding smooth rarefaction similarity solution is indicated that explains the collapse of this non-entropy shock wave. A survey on entropy–viscosity methods developed in the last fifty years is included.