TY - JOUR
T1 - Occurrence and non-appearance of shocks in fractal Burgers equations
AU - Alibaud, Nathael
AU - Droniou, Jerome
AU - Vovelle, Julien
PY - 2007
Y1 - 2007
N2 - We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently large initial conditions, by giving a result which states that, for smooth small initial data, the solution remains at least Lipschitz continuous.
AB - We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently large initial conditions, by giving a result which states that, for smooth small initial data, the solution remains at least Lipschitz continuous.
UR - http://www.worldscientific.com/doi/abs/10.1142/S0219891607001227
U2 - 10.1142/S0219891607001227
DO - 10.1142/S0219891607001227
M3 - Article
SN - 0219-8916
VL - 4
SP - 479
EP - 499
JO - Journal of Hyperbolic Differential Equations
JF - Journal of Hyperbolic Differential Equations
IS - 3
ER -