Vanishing Viscosity Research Articles

Existence of solutions to k‐Wave models of nonlinear ultrasound propagation in biological tissue

AbstractWe investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k‐Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Second, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well.

Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α≥5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1−2α,∞,p) and (2α/γ+1−2α,γ,∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff Hη⁎ measure, where η⁎>0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.

A KAM Approach to the Inviscid Limit for the 2D Navier–Stokes Equations

In this paper, we investigate the inviscid limit ν→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u \\rightarrow 0$$\\end{document} for time-quasi-periodic solutions of the incompressible Navier–Stokes equations on the two-dimensional torus T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}^2$$\\end{document}, with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier–Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order O(ν2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\ u ^2)$$\\end{document} and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier–Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems.

Characteristic boundary layers in the vanishing viscosity limit for the Hunter-Saxton equation

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method.

Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function

We study the Aw-Rascle system in a one-dimensional domain with periodic boundary conditions, where the offset function is replaced by the gradient of the function ρnγ, where γ→∞. The resulting system resembles the 1D pressureless compressible Navier-Stokes system with a vanishing viscosity coefficient in the momentum equation and can be used to model traffic and suspension flows. We first prove the existence of a unique global-in-time classical solution for fixed n. Unlike the previous result for this system, we obtain global existence without needing to add any approximation terms to the system. This is by virtue of a n-uniform lower bound on the density which is attained by carrying out a maximum-principle argument on a suitable potential, Wn=ρn−1∂xwn. Then we prove the convergence to a weak solution of a hybrid free-congested system as n→∞, which is known as the hard-congestion model.

Riemann problem for a 2times 2 hyperbolic system with time-gradually-degenerate damping

This paper is focused on the Riemann problem for a 2times 2 hyperbolic system of conservation laws with a time-gradually-degenerate damping. Two kinds of non-self-similar solutions involving the delta-shocks and vacuum are obtained using the variable substitution method. The generalized Rankine-Hugoniot relation and entropy condition are clarified for the delta-shock. Furthermore, the vanishing viscosity method proves the existence, uniqueness, and stability of non-self-similar solutions.

Zero-noise dynamics after collapse for three point vortices

The continuation of point vortex dynamics after a vortex collapse is investigated by means of a regularization procedure consisting in introducing a small stochastic diffusive term, that corresponds to a vanishing viscosity. In contrast with deterministic regularization, in which a cutoff interaction selects in the limit a single trajectory of the system after collapse, the zero-noise method produces a probability distribution supported by trajectories satisfying relevant conservation laws of the point vortex system.

Rate-independent stochastic evolution equations: Parametrized solutions

By extending to the stochastic setting the classical vanishing viscosity approach we prove the existence of suitably weak solutions of a class of nonlinear stochastic evolution equation of rate-independent type. Approximate solutions are obtained via viscous regularization. Upon properly rescaling time, these approximations converge to a parametrized martingale solution of the problem in rescaled time, where the rescaled noise is given by a general square-integrable cylindrical martingale with absolutely continuous quadratic variation. In absence of jumps, these are strong-in-time martingale solutions of the problem in the original, not rescaled time.

Which Measure-Valued Solutions of the Monoatomic Gas Equations are Generated by Weak Solutions?

Contrary to the incompressible case, not every measure-valued solution of the compressible Euler equations can be generated by weak solutions or a vanishing viscosity sequence. In the present paper we give sufficient conditions on an admissible measure-valued solution of the isentropic Euler system to be generated by weak solutions. As one of the crucial steps we prove a characterization result for generating {mathcal {A}}-free Young measures in terms of potential operators including uniform L^{infty }-bounds. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing that are also necessary conditions for generating a measure-valued solution by weak solutions or a vanishing viscosity sequence and will point out that the resulting gap mainly results from obtaining only uniform L^p-bounds for 1<p<infty instead of p=infty .

Impact of dissipation ratio on vanishing viscosity solutions of the Riemann problem for chemical flooding model

The solutions for a Riemann problem arising in chemical flooding models are studied using vanishing viscosity as an admissibility criterion. We show that when the flow function depends non-monotonically on the concentration of chemicals, non-classical undercompressive shocks appear. The monotonic dependence of the shock velocity on the ratio of dissipative coefficients is proven. For that purpose we provide the classification of the nullcline configurations for the traveling wave dynamical systems and analyze the saddle–saddle connections.

Existence of Optimal Control for Nonlinear Fokker–Planck Equations in \(\boldsymbol{L^1(\mathbb{R}^d)}\).

This work is concerned with the existence of optimal controllers for the Bolza optimal control problem governed by the nonlinear Fokker–Planck equation in with control input in the drift term. The solution to the control state system is a weak (mild) solution obtained from a vanishing viscosity approximation scheme. One obtains in particular the existence for the stochastic optimal control problem governed by McKean–Vlasov SDEs. For this problem, one proves the existence of a stochastic Markov optimal controller in feedback form.

Propagation dynamics of nonlocal dispersal monostable equations in time-space periodic habitats

This paper is concerned with the propagation dynamics of nonlocal dispersal monostable equations in time-space periodic habitats. We first show that such an equation admits a single spreading speed in every direction under certain conditions and then give several spreading properties in terms of spreading speeds such as asymptotic spreading ray speeds and asymptotic spreading sets. Furthermore, we consider the dependence of the spreading speed on the dispersal rate and reaction term and prove that taking the temporal average or spatial average can decrease the spreading speed. Finally, we employ the viscosity vanishing method to establish the existence of time-space periodic traveling fronts with the critical speed in every direction under the partially temporally homogeneous case and partially nearly flat case, which solves partially the open problem raised by Rawal, Shen, and Zhang (Discrete Contin. Dyn. Syst.35 (2015) 1609–1640).

Two-Dimensional Flow on the Sphere

Equilibrium statistical mechanics predicts that inviscid, two-dimensional, incompressible flow on the sphere eventually reaches a state in which spherical harmonic modes of degrees n=1 and n=2 hold all the energy. By a separate theory, such flow is static in a reference frame rotating at angular speed 2Ω/3 with respect to the inertial frame. The vorticity field in the static frame is an accident of the initial conditions, but, once established, it lasts forever under the stated assumptions. We investigate the possibility of such behavior with a stereographic-coordinate model that conserves energy and enstrophy when the viscosity vanishes.

Reinforcement learning based optimal synchronization control for multi-agent systems with input constraints using vanishing viscosity method

This paper investigates the optimal synchronization and disturbance rejection problem for leader-follower multi-agent systems (MASs) subject to constrained input using online actor-critic (A-C) algorithm. Different from the existing works on optimal synchronization problems, the common smoothness assumption on the value function used in the typically reinforcement learning (RL) formulations is not satisfied due to the input constraints. To relax this assumption, the vanishing viscosity method is introduced to construct a more general value function which makes the modified Hamilton–Jacobi-Isaacs (HJI) equation admit a smooth solution. Based on the modified HJI equation, an improved A-C neural networks (NNs) algorithm is developed to find a smooth approximation optimal control solution. Finally, a simulation is performed to demonstrate the effectiveness of the proposed approach.

The strong vanishing viscosity limit with Dirichlet boundary conditions

We adapt methodology of Tosio Kato to establish necessary and sufficient conditions for the solutions to the Navier–Stokes equations with Dirichlet boundary conditions to converge in a strong sense to a solution to the Euler equations in the presence of a boundary as the viscosity is taken to zero. We extend existing conditions for no-slip boundary conditions to allow for nonhomogeneous Dirichlet boundary conditions and curved boundaries, establishing several new conditions as well. We give a brief comparison of various correctors appearing in the literature used for similar purposes.

Vanishing Viscosity Limit for the 3D Incompressible Micropolar Equations in a Bounded Domain

In this paper, we investigate the vanishing viscosity limit of the 3D incompressible micropolar equations in bounded domains with boundary conditions. It is shown that there exist global weak solutions of the micropolar equations in a general bounded smooth domain. In particular, we establish the uniform estimate of the strong solutions for when the boundary is flat. Furthermore, we obtain the rate of convergence of viscosity solutions to the inviscid solutions as the viscosities tend to zero (i.e., (ε,χ,γ,κ) → 0).

Vanishing viscosity in mean-field optimal control

We show the existence of Lipschitz-in-space optimal controls for a class of mean-field control problems with dynamics given by a non-local continuity equation. The proof relies on a vanishing viscosity method: we prove the convergence of the same problem where a diffusion term is added, with a small viscosity parameter. By using stochastic optimal control, we first show the existence of a sequence of optimal controls for the problem with diffusion. We then build the optimizer of the original problem by letting the viscosity parameter go to zero.

Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches

We address the problem of deterministic sequential estimation for a nonsmooth dynamics governed by a variational inequality. An example of such dynamics is the Skorokhod problem with a reflective boundary condition. For smooth dynamics, Mortensen introduced in 1968 a nonlinear estimator based on likelihood maximisation. Then, starting with Hijab in 1980, several authors established a connection between Mortensen’s approach and the vanishing noise limit of the robust form of the so-called Zakai equation. In this paper, we investigate to what extent these methods can be developed for dynamics governed by a variational inequality. On the one hand, we address this problem by relaxing the inequality constraint by penalization: this yields an approximate Mortensen estimator relying on an approximating smooth dynamics. We verify that the equivalence between the deterministic and stochastic approaches holds through a vanishing noise limit. On the other hand, inspired by the smooth dynamics approach, we study the vanishing viscosity limit of the Hamilton-Jacobi equation satisfied by the Hopf-Cole transform of the solution of the robust Zakai equation. In contrast to the case of smooth dynamics, the zero-noise limit of the robust form of the Zakai equation cannot be understood in our case from the Bellman equation on the value function arising in Mortensen’s procedure. This unveils a violation of equivalence for dynamics governed by a variational inequality between the Mortensen approach and the low noise stochastic approach for nonsmooth dynamics.

Self‐similar viscosity approach to the Riemann problem for a strictly hyperbolic system of conservation laws

Here, we study the Riemann problem for a strictly hyperbolic system of conservation laws, which occurs in gas dynamics and nonlinear elasticity. We establish the existence and uniqueness of the solution of Riemann problem containing delta shock wave by employing self‐similar vanishing viscosity approach. We prove that delta shock is stable under self‐similar viscosity perturbation, which ensures that delta shock wave is a unique entropy solution.