The forced inviscid Burgers equation as a model for nonlinear interactions among dispersive waves

One-dimensional turbulence with Burgers

“…In particular, ∂ x φ has a jump at x = x except for specific values of p for which V( x) = 0. We also recall that the solutions of ( 6) are not unique, as emphasized in Remark 1 below and in [10,12]. Indeed, to each minimum x 0 of √ V(x) corresponds (at least) one viscosity solution.…”

“…On the other hand, the viscosity solution φ(x) to ( 6) is unique for |p| ≥ 2 √ 2/π (again, up to constants). We can also revisit these results at the level of the conservation law, following the computations in [12]. Indeed, ū = p + ∂ x φ is a stationary entropy solution of the Burgers equation ( 1).…”

“…We do not claim that the above result is new, and in fact, we will rely on previous results (especially concerning the long-time behavior or the homogenization of Hamilton-Jacobi equations) to prove Theorem 1. Following [12], we will also study the case when the forcing f is of the form f (x − ωt).…”

“…In this paper, we are interested in the long time behavior of the forced inviscid Burgers equation. Indeed, as advertised by Menzaque, Rosales, Tabak and Turner, this equation is the simplest good example simulating the mechanism of weak wave turbulence [12]: "the nonlinear term in (1) has two combined functions: to transfer energy among the various Fourier components of u and to dissipate energy at shocks. Thus the inertial cascade and the dissipation are modeled by the same term.…”

Nonlinear observer for distributed parameter systems described by decoupled advection equations

“…The system is modeled based on a distributed parameter model known as the advection equation. This class of equation includes the nonlinear transport equation/inviscid Burgers equation (Menzaque et al., 2001) and the dispersionless Korteweg–de Vries equation/Riemann equation (Choudhuri et al., 2007) where u = u ( t,x ) and ut=∂u∂t,ux=∂u∂x. Such equations have applications in modeling gas dynamics, water waves, flood waves in rivers, transport of pollutants, traffic flow, etc.…”