Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux I

We discuss different notions of continuous solutions to the balance law 02tu+ 02x(f(u))=gg bounded,f 08C2 extending previous works relative to the flux f(u)=u2. We establish the equivalence among distributional solutions and a suitable notion of Lagrangian solutions for general smooth fluxes. We eventually find that continuous solutions are Kruzkov iso-entropy solutions, which yields uniqueness for the Cauchy problem. We also reduce the ODE on any characteristics under the sharp assumption that the set of inflection points of the flux f is negligible. The correspondence of the source terms in the two settings is a matter of the companion work [2], where we include counterexamples when the negligibility on inflection points fails

On Nonlinear Stochastic Balance Laws

We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial $BV$ bounds for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in $L^1$ of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed

A Regularization of Burgers Equation using a Filtered Convective Velocity

This paper examines the properties of a regularization of Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as convectively filtered Burgers (CFB) equation. A physical motivation behind the filtering technique is presented. An existence and uniqueness theorem for multiple dimensions and a general class of filters is proven. Multiple invariants of motion are found for the CFB equation and are compared with those found in viscous and inviscid Burgers equation. Traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of inviscid Burgers equation with the correct wave speed. Accurate numerical simulations are conducted in 1D and 2D cases where the shock behavior, shock thickness, and kinetic energy decay are examined. Energy spectrum are also examined and are shown to be related to the smoothness of the solutions

Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation

We study the one-dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equationpartial derivative(t)u= epsilon partial derivative(2)(x)u - partial derivative(x)f(u) + f'(u).Metastable behaviour appears when the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches its steady state in a very long time interval. A rigorous analysis is used to study such behaviour by means of the construction of a one-parameter family {U-epsilon(x; xi)}(xi) of approximate stationary solutions and of a linearisation of the original system around an element of this family. We obtain a system consisting of an ODE for the parameter xi, describing the position of the interface coupled with a PDE for the perturbation v and defined as the difference v := u - U-epsilon. The key of our analysis are the spectral properties of the linearised operator around an element of the family {U-epsilon}: the presence of a first eigenvalue, small with respect to epsilon, leads to metastable behaviour when epsilon << 1