We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we establish that the diffusive-dispersive solutions are uniformly bounded in a space Lp ($p$ arbitrary large, depending on the nonlinearity of the diffusion) and converge to the classical, entropy solution of the corresponding multidimensional, hyperbolic conservation law. Previous results were restricted to one-dimensional equations and specific spaces Lp. Our proof is based on DiPerna's uniqueness theorem in the class of entropy measure-valued solutions.
Given a first-order nonlinear hyperbolic system of conservation laws endowed with a convex entropy-entropy flux pair, we can consider the class of weak solutions containing shock waves depending upon some small scale parameters. In this Note, we define and derive several classes of entropy-dissipating augmented models, as we call them, which involve (possibly nonlinear) second- and third-order augmentation terms. Such terms typically arise in continuum physics and model the viscosity and capillarity effects in a fluid, for instance. By introducing a new notion of positive entropy production th…
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws where the far field states are prescribed. We proved that the solution of the Cauchy problem tends toward the constant state as time goes to infinity.
We study the multi-dimensional Burgers equation $u_t + u u_{x_1} + \dots + u^d u_{x_d} = 0$. We prove that the $L^\infty$ norm of entropy solutions of this equation decays polynomially as $t \to \infty$ in terms of the $L^1$ norm of the initial data only.
This paper deals with the rigorous study of the diffusive stress relaxation in the multidimensional system arising in the mathematical modeling of viscoelastic materials. The control of an appropriate high order energy shall lead to the proof of that limit in Sobolev space. It is shown also as the same result can be obtained in terms of relative modulate energies.
The aim is to assess the combined effect of diffusion and dispersion on shocks in the moderate dispersion regime. For a diffusive dispersive approximation of the equations of one-dimensional elasticity (or p-system), we study convergence of traveling waves to shocks. The problem is recast as a Hamiltonian system with small friction, and an analysis of the length of oscillations yields convergence in the moderate dispersion regime $\eps, δ\to 0$ with $δ= o(\eps)$, under hypotheses that the limiting shock is admissible according to the Liu E-condition and is not a contact discontinuity at either…
We investigate the properties of traveling wave solutions to hyperbolic conservation laws augmented with diffusion and dispersion, and review the existence and qualitative properties of the associated kinetic functions, which characterize the class of admissible shock waves selected by such regularizations.
We present local existence and uniqueness results for the following $2+1$ dispersive diffusive equation due to P. Hall arising in modeling of river braiding: $$u_{yyt} - γu_{xxx} -αu_{yyyy} - βu_{yy} + \left (u^2 \right)_{xyy} = 0$$ for $(x,y) \in [0, 2π] \times [0, π]$, $t> 0$, with boundary condition $u_{y}=0=u_{yyy}$ at $y=0$ and $y=π$ and $2π$ periodicity in $x$, using a contraction mapping argument in a Bourgain-type space $T_{s,b}$. We also show that the energy $\| u \|^2_{L^2} $ and cumulative dissipation $\int_0^t \| u_y \|_{L^2}^2 dt$ are globally controlled in time.
We generalize the Abstract Interpolation Lemma proved by the authors in [2]. Using this extension, we show in a more general context, the persistence property for the generalized Korteweg-de Vries equation, see (1.2), in the weighted Sobolev space with low regularity in the weight. The method used can be applied for other nonlinear dispersive models, for instance the multidimensional nonlinear Schrodinger equation.
This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The a…
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d \ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and $L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and dif…
The critical Burgers equation $\partial_t u + u \partial_x u + Λu = 0$ is a toy model for the competition between transport and diffusion with regard to shock formation in fluids. It is well known that smooth initial data does not generate shocks in finite time. Less is known about the long-time behavior for `shock-like' initial data: $u_0 \to \pm a$ as $x \to \mp \infty$. We describe this long-time behavior in the general setting of multidimensional critical scalar conservation laws $\partial_t u + \text{div}f(u) + Λu = 0$ when the initial data has limits at infinity. The asymptotics are give…
