Weak dissipative solutions to a free-boundary problem for finitely extensible bead-spring chain molecules: Variable viscosity coefficients

In this paper, the Cauchy problem for nonstrictly hyperbolic systems of conservation laws modeling polymer flooding in enhanced oil recovery is studied, and a short proof of global existence of weak entropy solutions is obtained by using a new technique from the compensated compactness theorem coupled with the total variation estimate on one family of Riemann invariants. This work extends in some sense the previous works [T. Barkve, SIAM J. Appl. Math., 49 (1989), pp. 789--798; T. Johansen and R. Winther, SIAM J. Math. Anal., 19 (1988), pp. 541--566] about the global existence of weak solutions for the Riemann problem and by the author [J. Funct. Anal., 261 (2011), pp. 2797--2815] about the global weak solution of the Cauchy problem for an extended hyperbolic system of Keyfitz--Kranzer or Aw--Rascle type.

A note on global existence of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.

In this paper, we prove the global existence of weak solutions to one-dimensional compressible isentropic Navier–Stokes–Poisson equations with density-dependent viscosity and free boundaries. The initial density ρ0∊W1,2n is bounded below by a positive constant, and the initial velocity u0∊L2n. In contrast to Jiang et al. [“Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity,” Methods Appl. Anal. 12, 239 (2005)], the Sobolev exponent n is less in this paper, and the viscosity coefficient μ=μ(ρ) is a general function of ρ including the cases μ(ρ)=c0ρθ (0<θ<1) and μ(ρ)=c0, where c0 is a positive constant.

Global weak solutions to a spatio-temporal fractional Landau–Lifshitz–Bloch equation

We study the existence of global weak solutions for Cauchy-Dirichlet problem for evolutional p-Laplacian systems. The initial data is belonging to the Sobolev space W1,p(Ω, ℝn). We use a varitional (like) method to treat the existence of a weak solution, this method seems somehow simply than the Galerkin method.

We propose a new notion of weak solutions (dissipative solutions) for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. As an application we prove the strong convergence of a general relaxation-type approximation for such equations.

We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions.

Consider a smooth bounded domain \Omega\subseteq\mathbb R^3 with boundary \partial\Omega , a time interval [0,T) , with T\in(0,\infty] , and the Navier–Stokes system in [0,T) \times \Omega , with initial value u_0 \in L^2_{\sigma} (\Omega) and external force f= {\mathrm{div}}\,F , F \in L^2 (0,T;L^2(\Omega)) . Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u_{\vert{\partial \Omega}}=0 , {\mathrm{div}}\,u=0 to the more general class of Leray-Hopf type weak solutions u with general data u_{\vert{\partial \Omega}} =g , {\mathrm{div}}\,u=k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u=v+E with some vector field E in [0,T)\times \Omega satisfying the (linear) Stokes system with f=0 and nonhomogeneous data. This reduces the general system to a perturbed Navier–Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier–Stokes system we get the existence of global weak solutions for the more general system.

We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluid and the momentum equations for the mixture. This model can be derived from the compressible two-fluid model with equal velocities (Bresch et al., in Arch Rational Mech Anal 196:599–629, 2010) and from a scaling limit of the Vlasov-Fokker-Planck/compressible Navier-Stokes (Mellet and Vasseur, Commun Math Phys 281(3):573–596, 2008) (see also the compressible Oldroyd-B model with stress diffusion (Barrett et al., Commun Math Sci 15:1265–1323, 2017). Another interesting connection is that it is formally the equations of compressible magnetohydrodynamic (MHD) flows without resistivity in two dimensions under the action of vertical magnetic field (Li and Sun, J Differ Equ 267(6): 3827–3851, 2019). Under weak assumptions on the initial data which can be discontinuous, unbounded and large as well as involve transition to pure single-phase points or regions, we show existence of global weak solutions with finite energy. The essential novelty of this work, compared with previous works on the same model, is that transition to each single-phase flow is allowed without any constraints between adiabatic constants or two densities. It means that one of the phases can vanish in a point while the other can persist. The lack of enough regularity for each densities brings up essential difficulties in the two-component pressure compared with the single-phase model, i.e., compressible Navier-Stokes equations. The key points to achieve the main result rely on the variables reduction technique for the pressure function, domain separation, and some new estimates. As a byproduct, we obtain the existence of global weak solutions to the compressible MHD system without resistivity in two dimensions under the action of non-negatively vertical magnetic field, which represents a step forward to the study of the global large solution to the compressible MHD system without resistivity.

The initial value problem to the multi-dimensional drift-flux model for two-phase flow is investigated in this paper, and the global existence of weak solutions with finite energy is established for general pressure-density functions without the monotonicity assumption.

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.