IJPAM: Volume 37, No. 4 (2007)

$L^1$-STABILITY OF CONSTANTS IN
MULTI-DIMENSIONAL CONSERVATION
LAWS WITH VISCOSITY

Rita Cavazzoni
Facoltá di Ingegneria - Sede di Modena
Universitá Degli Studi di Modena e Reggio Emilia
Via Vignolese 905, Modena, 41100, ITALY
e-mail: [email protected]


Abstract.We study a class of multi-dimensional scalar viscous conservation laws. The solution of the Cauchy problem satisfies comparison principles, $L^1$-contraction property and preserves the total mass. After proving $L^p$-contraction ($p>1$) of the solution and an $L^2$-decay estimate for the gradient, we state the $L^1$-stability of constant states, in the case where the initial disturbance of the initial value with respect to the fixed constant belongs to $L^1$ and has zero mass. Similarly to in one-dimensional problems, the $L^1$-stability of constants implies the $L^1$-stability of shock waves.

Received: April 5, 2007

AMS Subject Classification: 35L65, 35B40, 35K15

Key Words and Phrases: viscous conservation laws, Cauchy problem, $L^1$-stability, shock waves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 37
Issue: 4