Related papers: A Uniqueness Condition for Conservation Laws with …
Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds,…
Dissipation anomaly-the persistence of finite energy dissipation in the inviscid limit-is a hallmark of turbulence, sometimes regarded as the "zeroth law" of turbulent flows. Here, we demonstrate that this phenomenon is not exclusive to…
We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schr\"odinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the…
We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction…
In this paper, we consider scalar conservation laws with smoothly varying spatially heterogeneous flux that is convex in the conserved variable. We show that under certain assumptions, a shock wave connecting two constant states emerges in…
We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function…
We consider cross-diffusion systems describing evolution of two species $u$ and $v$ moving according to Darcy's law with the pressure law $p(s) = \frac{1}{\alpha-1} s^{\alpha-1}$ where $s=u+v$. One of the most challenging questions in the…
In this paper, we discuss the asymptotic behaviour of the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear Laplacian viscosity. Firstly, we obtain the existence, uniqueness and regularity of…
We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an $H(\mathrm{div})$-conforming space,…
We study the equation of one-dimensional quasistatic nonlinear viscoelasticity with Dirichlet boundary conditions, in the particular case that the underlying dissipation geometry (provided by the viscosity) is comparable to the Bhattacharya…
We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier-Stokes system. The two fluids are assumed to be immiscible, meaning that they are separated by an interface. With…
This article deals with the regularity of the entropy solutions of scalar conservation laws with discontinuous flux. It is well-known [Adimurthi et al., Comm. Pure Appl. Math. 2011] that the entropy solution for such equation does not admit…
This paper concerns preservation of velocity and pressure equilibria in smooth, compressible, multicomponent flows in the inviscid limit. First, we derive the velocity-equilibrium and pressure-equilibrium conditions of a standard…
In a recent paper [F. Vega Reyes et al., Phys. Rev. Lett. 104, 028001 (2010)] we presented a preliminary description of a special class of steady Couette flows in dilute granular gases. In all flows of this class the viscous heating is…
In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$…
We are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally…
A variety of real-world applications are modeled via hyperbolic conservation laws. To account for uncertainties or insufficient measurements, random coefficients may be incorporated. These random fields may depend discontinuously on the…
We consider the generalized Leray's problem with the Navier-slip boundary condition in an infinite pipe $\mathcal{D}=\Sigma\times\mathbb{R}$. We show that if the flux $\Phi$ of the solution is no larger than a critical value that is…
A complex integral formula provides an explicit solution of the initial value problem for the nonlinear scala 1D equation $u_t+[f(u)]_x = 0$, for any flux $f(u)$ and initial condition $u_0(x)$ that are analytic. This formula is valid at…
This article concerns a scalar multidimensional conservation law where the flux is of Panov type and may contain spatial discontinuities. We define a notion of entropy solution and prove that entropy solutions are unique. We propose a…