Existence of finite energy weak solutions for the equations MHD of compressible fluids

This article concerns the existence of global weak solutions for a compressible Magnetohydrodynamic model. We assume the viscosity and the resistivity to be constant and we prove that Feireisl and Lions's strategies dedicated to the usual barotropic compressible flows may be extended to our system. The only difficulty to be taken into account is the magnetic field dependency. The case with density-dependent viscosity and resistivity coefficients will be treated in a forthcoming paper following Bresch and Desjardins's strategy.     

Symmetry analysis of a system of modified shallow-water equations

We revise the symmetry analysis of a modified system of one-dimensional shallow-water equations (MSWE) recently considered by Raja Sekhar and Sharma [Commun Nonlinear Sci Numer Simulat 2012;20:630–36]. Only a finite dimensional subalgebra of the maximal Lie invariance algebra of the MSWE, which in fact is infinite dimensional, was found in the aforementioned paper. The MSWE can be linearized using a hodograph transformation. An optimal list of inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra is constructed and used for Lie reductions. Non-Lie solutions are found from solutions of the linearized MSWE.     

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.     

Some exact solutions of the ideal MHD equations through symmetry reduction method

We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3+1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1?r?4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.     

Invariant solutions of one-dimensional equations of two-temperature relaxation gas dynamics

The group analysis method is applied to the plane one-dimensional equations of two-temperature gas dynamics. The complete classification of the equations with respect to admitted Lie group is studied. All invariant solutions are analyzed and their comparisons with the known invariant solutions of the ideal gas dynamics system are presented.     

Global weak solutions to a class of compressible non-Newtonian fluids with vacuum

The global weak solution of an initial-boundary value problem for a compressible non-Newtonian fluid is studied in a three-dimensional bounded domain. By the techniques of artificial pressure, a solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global weak solution to the three-dimensional compressible non-Newtonian fluid with vacuum and large data is established.     

A blow-up criterion for classical solutions to the compressible Navier-Stokes equations

In this paper, we obtain a blow-up criterion for classical solutions to the 3-D compressible Navier-Stokes equations just in terms of the gradient of the velocity, analogous to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, the initial vacuum is allowed in our case.     

Stress and heat flux stabilization for viscous compressible medium equations with a nonmonotone state function

We study a system of quasilinear equations describing one-dimensional flow of a viscous compressible heat-conducting medium with a nonmonotone state function and mass force. The large-time behavior of solutions is considered for arbitrarily large initial data. In spite of possible nonuniqueness and discontinuity of the stationary solution, we prove L²-stabilization for the stress and heat flux as t → ∞ along with corresponding global energy estimates for them. The new method of proof utilizes a combination of energy type equalities for the stress and heat flux. Consequently, H¹-stabilization of the velocity and temperature along with global estimates for their derivatives are valid as well.     

Self-similar solutions to the isothermal compressible Navier-Stokes equations

^** Email: guo_zhenhua{at}iapcm.ac.cn^*** Email: jiang{at}iapcm.ac.cn We investigate the self-similar solutions to the isothermalcompressible Navier–Stokes equations. The aim of thispaper is to show that there exist neither forward nor backwardself-similar solutions with finite total energy. This generalizesthe results for the incompressible case in Neas, J., Rika, M.& verák, V. (1996, On Leray's self-similar solutionsof the Navier-Stokes equations. Acta. Math., 176, 283–294),and is consistent with the (unproved) existence of regular solutionsglobally in time for the compressible Navier–Stokes equations.     

Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.     

Symmetries and similarity solutions for the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane

We perform a complete analysis of all the Lie point symmetries admitted by the equation describing the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane. We then investigate the existence of group-invariant solutions for particular values of the power-law parameter β.     

Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ₀, u ₀ satisfy the compatibility condition for some radially symmetric g ∈ L². The initial density ρ₀ needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state

We consider the Navier-Stokes system describing motions of viscous compressible heat-conducting and “self-gravitating” media. We use the state function of the form p(u,θ)=p₀(u)+p₁(u)θ linear with respect to the temperature θ, but we admit rather general nonmonotone functions p₀ and p₁ of u, which allows us to treat various physical models of nuclear fluids (for which p and u are the pressure and the specific volume) or thermoviscoelastic solids. For solutions to an associated initial-boundary value problem with “fixed-free” boundary conditions and arbitrarily large data, we prove a collection of estimates independent of time interval, including uniform two-sided bounds for u, and describe asymptotic behavior as t→∞. Namely, we establish the stabilization pointwisely and in L^q for u, in L² for θ, and in L^q for v (the velocity), for any q∈[2,∞). For completeness, the proof of the corresponding global existence theorem is also included.     

Global strong solutions for a class of compressible non‐Newtonian fluids with vacuum

We study a class of compressible non‐Newtonian fluids in one space dimension. We prove, by using iterative method, the global time existence and uniqueness of strong solutions provided that the initial data satisfy a compatibility condition and the initial density is small in its H¹‐norm. The main difficulty is due to the strong nonlinearity of the system and the initial vacuum. Copyright © 2010 John Wiley & Sons, Ltd.     

A note on a symmetry analysis and exact solutions of a nonlinear fin equation

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.     

A new family of polynomial solutions for charged fluid spheres

We present a new family of relativistic charged fluid spheres described by a space time metric with a prescribed metric potential g₄₄ along with the specific choice of electric field intensity which vanishes at the centre and monotonically increasing towards the pressure free surface. The charged fluid solutions so obtained satisfy all the energy conditions along with the velocity of sound which decreases away from the centre. Also the solutions are utilized to construct the models of super-dense stars.     

On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion

The linearized initial boundary value problem describing the motion of the viscous compressible fluid is studied under Dirichlet zero condition in bounded and unbounded domains. The resolvent estimate for the corresponding operator is proved in the L_p framework and the sharp inner estimate of the resolvent set is obtained. Copyright © 2004 John Wiley Sons, Ltd.     

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n>1 dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.