scholarly journals Exponential convergence of solutions for random Hamilton–Jacobi equations

2019 ◽  
Vol 8 (3) ◽  
pp. 544-579
Author(s):  
Renato Iturriaga ◽  
Konstantin Khanin ◽  
Ke Zhang
Keyword(s):  
Exponential Convergence ◽  
Convergence Of Solutions ◽  
Jacobi Equations

scholarly journals Exponential convergence to the Maxwell distribution of solutions of spatially inhomogeneous Boltzmann equations

2019 ◽  
Vol 32 (01) ◽  
pp. 2050001
Author(s):  
Zhou Gang
Keyword(s):  
Rate Of Convergence ◽  
Hard Sphere ◽  
Weighted Space ◽  
Exponential Convergence ◽  
Maxwell Distribution ◽  
Boltzmann Equations ◽  
Convergence Of Solutions ◽  
Spatially Inhomogeneous ◽  
Different Temperatures ◽  
Spatially Homogeneous

We consider the rate of convergence of solutions of spatially inhomogeneous Boltzmann equations, with hard-sphere potentials, to some equilibriums, called Maxwellians. Maxwellians are spatially homogeneous static Maxwell velocity distributions with different temperatures and mean velocities. We study solutions in weighted space [Formula: see text]. The result is that, assuming the solution is sufficiently localized and sufficiently smooth, then the solution, in [Formula: see text]-space, converges to a Maxwellian, exponentially fast in time.

scholarly journals Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Qinbo Chen
Keyword(s):  
Contact Problem ◽  
Viscosity Solution ◽  
Unknown Function ◽  
Jacobi Equation ◽  
Specific Solution ◽  
Convergence Of Solutions ◽  
Limit Solution ◽  
Peierls Barrier ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

Abstract Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Let H ⁢ ( x , p , u ) {H(x,p,u)} be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter λ > 0 {\lambda>0} , we denote by u λ {u^{\lambda}} the unique viscosity solution of the Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , λ ⁢ u ⁢ ( x ) ) = c . H\big{(}x,Du(x),\lambda u(x)\big{)}=c. Under quite general assumptions, we prove that u λ {u^{\lambda}} converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , 0 ) = c {H(x,Du(x),0)=c} . We also characterize the limit solution in terms of Peierls barrier and Mather measures.

scholarly journals Infinitesimal Isometries Along Curves and Generalized Jacobi Equations

2011 ◽  
Vol 23 (1) ◽  
pp. 377-394
Author(s):  
R. L. Foote ◽  
C. K. Han ◽  
J. W. Oh
Keyword(s):  
Jacobi Equations

Exponential convergence in entropy and Wasserstein for McKean–Vlasov SDEs

2021 ◽  
Vol 206 ◽  
pp. 112259
Author(s):  
Panpan Ren ◽  
Feng-Yu Wang
Keyword(s):  
Exponential Convergence

scholarly journals A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Timon S. Gutleb
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Jacobi Polynomial ◽  
Numerical Experiments ◽  
Exponential Convergence ◽  
Analytic Solutions ◽  
Convolution Type ◽  
Volterra Equations ◽  
Polynomial Bases ◽  
Sparsity Structure

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

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