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Register a new userViscous Conservation Laws in 1d With Measure Initial Data
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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.
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We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the decay of the solution as time increases, in relation with the nonlinearity. The main ingredient is our recent theory of divergence-free positive symmetric tensor. We apply in particular the so-called compensated integrability to a tensor which generalizes the one that L. Tartar used in one space dimension. It allows us to establish a Strichartz-like inequality, in a quasilinear context. This program is carried out in details for a multi-dimensional version of the Burgers equation.
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.
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 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.
We show the convergence of the zero relaxation limit in systems of $2 times 2$ hyperbolic conservation laws with stochastic initial data. Precisely, solutions converge to a solution of the local equilibrium approximation as the relaxation time tends to zero. The initial data are assumed to depend on finitely many random variables, and the convergence is then proved via the appropriate analogues of the compensated compactness methods used in treating the deterministic case. We also demonstrate the validity of this limit in the case of the semi-linear $p$-system; the well-posedness of both the system and its equilibrium approximation are proved, and the convergence is shown with no a priori conditions on solutions. This model serves as a prototype for understanding how asymptotic approximations can be used as control variates for hyperbolic balance laws with uncertainty.
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