## Bernoulli

• Bernoulli
• Volume 11, Number 4 (2005), 689-714.

### Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws

#### Abstract

We are interested in the one-dimensional scalar conservation law $\partial_t u(t,x)=\nu D^\alpha u(t,x)-\partial_xA(u(t,x))$ with fractional viscosity operator Dαv(x) = F-1(|ξ|αF(v)(ξ))(x) is the cumulative distribution function of a signed measure on R. We associate a nonlinear martingale problem with the Fokker-Planck equation obtained by spatial differentiation of the conservation law. After checking uniqueness for both the conservation law and the martingale problem, we prove existence thanks to a propagation-of-chaos result for systems of interacting particles with fixed intensity of jumps related to ν. The empirical cumulative distribution functions of the particles converge to the solution of the conservation law. As a consequence, it is possible to approximate this solution numerically by simulating the stochastic differential equation which gives the evolution of particles. Finally, when the intensity of jumps vanishes (ν→0) as the number of particles tends to +∞, we obtain that the empirical cumulative distribution functions converge to the unique entropy solution of the inviscid (ν=0) conservation law.

#### Article information

Source
Bernoulli, Volume 11, Number 4 (2005), 689-714.

Dates
First available in Project Euclid: 7 September 2005

https://projecteuclid.org/euclid.bj/1126126765

Digital Object Identifier
doi:10.3150/bj/1126126765

Mathematical Reviews number (MathSciNet)
MR2158256

Zentralblatt MATH identifier
1122.60063

#### Citation

Jourdain, Benjamin; Méléard, Sylvie; Woyczynski, Wojbor A. Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws. Bernoulli 11 (2005), no. 4, 689--714. doi:10.3150/bj/1126126765. https://projecteuclid.org/euclid.bj/1126126765

#### References

• [1] Biler, P. and Woyczynski, W.A. (1998) Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math., 59, 845-869.
• [2] Biler, P., Karch, G. and Woyczynski, W.A. (2001) Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire, 18, 613-637.
• [3] Bossy, M. and Talay, D. (1996) Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation. Ann. Appl. Probab., 6, 818-861.
• [4] Bossy, M. and Talay, D. (1997) A stochastic particle method for the McKean-Vlasov and the Burgers equation. Math. Comput., 66, 157-192.
• [5] Brezis, H. (1983) Analyse fonctionnelle. Paris: Masson.
• [6] Ethier, S.N. and Kurtz, T.G. (1986) Markov Processes. New York: Wiley.
• [7] Droniou, J. (2003) Vanishing non-local regularization of a scalar conservation law. Electron. J. Differential Equations, 2003(117).
• [8] Jacod, J. and Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag.
• [9] Jourdain, B. (2000) Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab., 2(1), 69-91.
• [10] Jourdain, B. (2002) Probabilistic characteristics method for a one-dimensional inviscid scalar conservation laws. Ann. Appl. Probab., 12, 334-360.
• [11] Jourdain, B., Méléard, S. and Woyczynski, W.A. (2005) A probabilistic approach for nonlinear equations involving the fractional Laplacian and a singular operator. Preprint CERMICS 2003- 254. Potential Anal., 23(1), 55-81.
• [12] Kellog G.L. (1994) Direct observation of substitutional-atom trapping on a metal surface. Phys. Rev. Lett., 72, 1662-1665.
• [13] Klafter J., Zumofen G., Shlesinger M.F. (1995) Lévy description of anomalous diffusion in dynamical systems. In M.F. Shlesinger, G.M. Zaslavsky and U. Frisch (eds), Lévy Flights and Related Topics in Physics, Lecture Notes in Phys. 450, pp. 196-215. Berlin: Springer-Verlag.
• [14] Komatsu, T. (1984) Pseudo-differential operators and Markov processes. J. Math. Soc. Japan, 36, 387-418.
• [15] Kruzhkov, S.N. (1970) First order quasilinear equations in several independent variables. Math. USSR, Sb., 10, 217-243.
• [16] Mann, J.A. and Woyczynski, W.A. (2001) Growing fractal interfaces in the presence of self-similar hopping surface diffusion. Phys. A, 291, 159-183.
• [17] Piryatinska A., Saichev A.I. and Woyczynski W.A. (2005) Models of anomalous diffusion: the subdiffusive case. Phys. A., 349, 375-420.
• [18] Saichev A.I. and Zaslavsky G.M. (1997) Fractional kinetic equations: solutions and applications. Chaos, 7, 753-764.
• [19] Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press.
• [20] Serre, D. (1996) Systèmes de lois de conservation. I: Hyperbolicité, entropies, ondes de choc. Paris: Diderot.