Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLocal conservation laws of second-order evolution equations
184
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78, 531-676], we completely describe local conservation laws of second-order (1+1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws.
rate research
Read More
We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator with several local conservation laws (local density, momentum and energy). We exhibit all equilibria and entropy-maximizing special modes, and we prove asymptotic exponential convergence of solutions to them with quantitative rate. This is the first complete picture of hypocoercivity and quantitative $H$-theorem for inhomogeneous kinetic equations in this setting.
The one-dimensional viscous conservation law is considered on the whole line $$ u_t + f(u)_x=eps u_{xx},quad (x,t)inRRtimesoverline{RP},quad eps>0, $$ subject to positive measure initial data. The flux $fin C^1(RR)$ is assumed to satisfy a $p-$condition, a weak form of convexity. Existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for viscous Hamilton-Jacobi equations.
The direct method based on the definition of conserved currents of a system of differential equations is applied to compute the space of conservation laws of the (1+1)-dimensional wave equation in the light-cone coordinates. Then Noethers theorem yields the space of variational symmetries of the corresponding functional. The results are also presented for the standard space-time form of the wave equation.
We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^{alpha} in Lambda^2(W)$ such that [ phi_{beta gamma}A^{beta}wedge A^{gamma}=0, ] for some non-degenerate symmetric $phi$.
We investigate integrability of Euler-Lagrange equations associated with 2D second-order Lagrangians of the form begin{equation*} int f(u_{xx},u_{xy},u_{yy}) dxdy. end{equation*} By deriving integrability conditions for the Lagrangian density $f$, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
