Leray–Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress–strain relation

We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba–Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.     

Global existence of classical solutions for the 3D generalized compressible Oldroyd-B model

In this paper, we prove the global existence of small classical solutions to the 3D generalized compressible Oldroyd-B system. It can be seen as compressible Euler equations coupling the evolution of stress tensor τ. The result mainly shows that singularity of solutions to compressible Euler equations can be prevented by the coupling of viscoelastic stress tensor. Moreover, unlike most complex fluids containing compressible Euler equations, the irrotational condition ∇×u=0 would not be proposed here to achieve the global well-posedness.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE

We consider the quantum Navier-Stokes equations for the viscous, compressible, heat conducting fluids on the three-dimensional torus T³. The model is based on a system which is derived by Jungel, Matthes and Milisic [15]. We made some adjustment about the relation of the viscosities of quantum terms. The viscosities and the heat conductivity coefficient are allowed to depend on the density, and may vanish on the vacuum. By several levels of approximation we prove the global-in-time existence of weak solutions for the large initial data.     

Existence of global solutions to multidimensional equations for Bingham fluids

We consider equations describing the multidimensional motion of compressible viscous (non-Newtonian) Bingham-type fluids, i.e., fluids with multivalued function relating the stresses to the tensor of strain rates. We prove the global existence theorem in time and in the initial data for the first initial boundary-value problem corresponding to flows in a bounded domain in the class of “weak” generalized solutions. In this case, we admit an anisotropic relation between the stress and strain rate tensors and study admissible relations of this kind in detail.     

Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism

In this paper, we prove the global existence of small smooth solutions to the three-dimensional incompressible Oldroyd-B model without damping on the stress tensor. The main difficulty is the lack of full dissipation in stress tensor. To overcome it, we construct some time-weighted energies based on the special coupled structure of system. Such type energies show the partial dissipation of stress tensor and the strongly full dissipation of velocity. In the view of treating “nonlinear term” as a “linear term”, we also apply this result to 3D incompressible viscoelastic system with Hookean elasticity and then prove the global existence of small solutions without the physical assumption (div–curl structure) as previous works.     

Existence of locally momentum conserving solutions to a model for heat-conducting flow

We prove the global-in-time existence of weak solutions to a model for a one-dimensional, viscous, compressible, heat-conducting fluid initially occupying a general open subset of an interval of finite length. The fluid equations are applied only on the support of the density, and this support is tracked in time. Accommodation must be made for infinitely many collisions of fluid packets occurring on a possibly dense set of times. Our approach avoids certain nonphysical behavior known to occur in solutions constructed as limits of solutions with artificial mass.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE THREE-DIMENSIONAL FULL COMPRESSIBLE QUANTUM EQUATIONS

We consider the quantum Navier-Stokes equations for the viscous, compressible, heat conducting fluids on the three-dimensional torus T~3. The model is based on a system which is derived by Jungel, Matthes and Milisic [15]. We made some adjustment about the relation of the viscosities of quantum terms.The viscosities and the heat conductivity coefficient are allowed to depend on the density, and may vanish on the vacuum. By several levels of approximation we prove the global-in-time existence of weak solutions for the large initial data.     

A note on barotropic compressible quantum Navier-Stokes equations

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations has been proved very recently, by Jüngel (2009) [1], if the viscosity constant is smaller than the scaled Plank constant. This paper extends the results to the case that the viscosity constant equals the scaled Plank constant. By using a new estimate on the square root of the solution, apparently not available in [1], the semiclassical limit for the viscous quantum Euler equations (which are equivalent to the barotropic compressible quantum Navier-Stokes equations) can be performed; then the results of this paper are obtained easily.     

A remark on weak solutions to the barotropic compressible quantum Navier–Stokes equations

In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained.     

Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case

We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.     

Global Classical Large Solutions with Vacuum to 1D Compressible MHD with Zero Resistivity

We establish the global-in-time existence and uniqueness of classical solutions of the initial boundary value problem for 1D compressible magnetohydrodynamics with zero magnetic resistivity, where the initial vacuum is allowed. In present paper, we do not require initial value to be small.     

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.     

On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids

The purpose of this work is to investigate the problem of global in time existence of sequences of weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. A class of density and temperature dependent viscosity and conductivity coefficients is considered. This result extends P.-L. Lions' work in 1993 [P.-L. Lions, Compacité des solutions des équations de Navier–Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, Sér. I 317 (1993) 115–120] restricted to barotropic flows, and provides weak solutions “à la Leray” to the full compressible model that includes internal energy evolution equation with thermal conduction effects. A partial answer is therefore given to this currently widely open problem, described for instance in P.-L. Lions' book [P.-L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998]. The proof uses the generalization to the temperature dependent case, of a new mathematical entropy equality derived by the authors in [D. Bresch, B. Desjardins, Some diffusive capillary models of Korteweg type, C. R. Acad. Sci., Paris, Section Mécanique 332 (11) (2004) 881–886]. The construction scheme of approximate solutions, using on additional regularizing effects such as capillarity, is provided in [D. Bresch, B. Desjardins, On the construction of approximate solutions for 2D viscous shallow water model and for compressible Navier–Stokes models, J. Math. Pures Appl. 86 (4) (2006) 362–368], and allows to use the stability arguments of this paper.     

Existence of global weak solutions for a 3D Navier–Stokes–Poisson–Korteweg equations

The purpose of this work is to study the global-in-time existence of weak solutions of a viscous capillary model of plasma expressed as a so-called Navier–Stokes–Poisson–Korteweg model for large data in three-dimensional space. Using the compactness argument, we prove the existence of global weak solutions in the classical sense to such system with a cold pressure.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities

We study the existence, uniqueness and asymptotic behavior, as well as the stability of a special kind of traveling wave solutions for competitive PDE systems involving intrinsic growth, competition, crowding effects and diffusion. The traveling waves are exclusive in the sense that as the variable goes to positive or negative infinity, different species are close to extinction or carrying capacity. We perform an appropriate affine transformation of the traveling wave equations into monotone form and construct appropriate upper and lower solutions. By this means, we reduce the existence proof to application of well-known theory about monotone traveling wave systems (cf. [A. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, MIA, Kluwer, Boston, 1989; J. Wu, X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651-687] and [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994]). Then, by using spectral analysis of the linearization over the profile, we prove the orbital stability of the traveling wave in some Banach spaces with exponentially weighted norm. Furthermore, we show that the introduction of some weight is necessary in the sense that, in general, traveling wave solutions with initial perturbations in the (unweighted) space C₀ are unstable (cf. [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994] and [D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981]).     

Global solutions of the Navier–Stokes equations for isentropic flow with large external potential force

We prove the global-in-time existence of weak solutions to the Navier–Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent γ ≥ 1. Initial data and solutions are small in L ² around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when γ = 1. A great deal of information about partial regularity and large-time behavior is obtained.     

Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds

We prove the global-in-time Strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds. We obtain estimates for the full set of wave admissible indices, including the endpoint. The key points are the properties of the microlocalized spectral measure of Laplacian on this setting showed in [18] and a Littlewood–Paley squarefunction estimate. As applications, we prove the global existence and scattering for a family of nonlinear wave equations on this setting.     

Global-in-time Hölder continuity of the velocity gradients for fluids with shear-dependent viscosities

For an evolutionary nonlinear fluid model characterized by the viscosity being a decreasing function of the modulus of the symmetric velocity gradient we establish the global-in-time existence of the solution with the H?lder continuous velocity gradients. Such a solution is unique in the class of weak solutions. We deal with the two dimensional space periodic problem.