Nonlocal systems of balance laws in several space dimensions with applications to laser technology

Abstract

For a class of systems of nonlinear and nonlocal balance laws in several space dimensions, we prove the local in time existence of solutions and their continuous dependence on the initial datum. The choice of this class is motivated by a new model devoted to the description of a metal plate being cut by a laser beam. Using realistic parameters, solutions to this model obtained through numerical integrations meet qualitative properties of real cuts. Moreover, the class of equations considered comprises a model describing the dynamics of solid particles along a conveyor belt.

Discontinuous nonlocal conservation laws and related discontinuous ODEs -- Existence, Uniqueness, Stability and Regularity

“…with Y (x 0 , x0 , v, L), X(x 0 , v, λ) as in Eq. (33). The proof consists of improving the estimate Eq.…”

An Age and Space Structured SIR Model Describing the Covid-19 Pandemic

“…For these kind of partial differential equations, a general well posedness theory is still missing. However, the different equations are coupled through the source terms, similarly to the cases considered in [9, 8] where well posedness is obtained, as well as the stability with respect to the parameters defining the equation, see [17].…”

An age and space structured SIR model describing the Covid-19 pandemic

“…For these kind of partial differential equations, a general well posedness theory is still missing. However, the different equations are coupled through the source terms, similarly to the cases considered in [8,15] where well posedness is obtained, as well as the stability with respect to the parameters defining the equation, see [16].…”

“…From left to right, the S, I, H and R populations Figure 16 Contour plots of the solution to (8) at time, above, t = 4.71 and, below, t = 4.72, roughly corresponding to 5 p.m. of the 4 th day. Above, the case with the commuters' movement (15), below the case with no movement. Apart from the spreading of the virus in the bottom left corner, the S distribution in the line above is symmetric w.r.t.…”