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The non-isentropic Euler system with periodic initial data in \({{\mathbb{R}}^1}\) is studied by analyzing wave interactions in a framework of specially chosen Riemann invariants, generalizing Glimm’s functionals and applying the method of approximate conservation laws and approximate characteristics. An \({{\mathcal O}(\varepsilon^{-2})}\) lower bound is established for the life span of the entropy solutions with initial data that possess \({\varepsilon}\) variation in each period.

In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3 for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on the other one; (iv) a new one, which characterizes the maximal existence time of the strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.

This paper concerns the well-posedness theory of the motion of physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in \cite{10',7,16'} by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary., Comment: To appear in Arch. Rational Mech. Anal

In this paper, we study the global well-posedness of the 2D compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ) = ρ β with β > 3, then the 2D compressible Navier–Stokes equations with the periodic boundary conditions on the torus $${\mathbb{T}^2}$$ admit a unique global classical solution (ρ, u) which may contain vacuums in an open set of $${\mathbb{T}^2}$$ . Note that the initial data can be arbitrarily large to contain vacuum states.

In this paper, the existence, uniqueness, and far field behavior of a class of subsonic flows with large vorticity for the steady Euler equations in infinitely long nozzles are established. More precisely, for any given convex horizontal velocity of incoming flow in the upstream, there exists a critical value m cr , if the mass flux is larger than m cr , then there exists a unique smooth subsonic Euler flow through the infinitely long nozzle. This well-posedness result is proved by a new observation for the method developed in Xie and Xin (SIAM J Math Anal 42:751–784, 2010) to deal with the Euler system. Furthermore, the optimal convergence rates of the subsonic flows at far fields are obtained via the maximum principle and an elaborate choice of the comparison functions.

In this paper, we characterize a set of physically acceptable boundary conditions that ensure the existence and uniqueness of a subsonic irrotational flow in a finitely long flat nozzle. Our results show that if the incoming flow is horizontal at the inlet and an appropriate pressure is prescribed at the exit, then there exist two positive constants m 0 and m 1 with m 0

In this paper, we study the global well-posedness of classical solution to 2D Cauchy problem of the compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ ( ρ ) = ρ β with β > 3 , then the 2D Cauchy problem of the compressible Navier–Stokes equations on the whole space R 2 admits a unique global classical solution ( ρ , u ) which may contain vacuums in an open set of R 2 . Note that the initial data can be arbitrarily large to contain vacuum states. Various weighted estimates of the density and velocity are obtained in this paper and these self-contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow.

In this paper, we investigate the vanishing viscosity limit problem for the 3-dimensional (3D) incompressible Navier–Stokes equations in a general bounded smooth domain of R3 with the generalized Navier-slip boundary conditions \(u^{\varepsilon}\cdot n = 0,\ n\times(\omega^{\varepsilon}) = [B u^{\varepsilon}]_{\tau}\ {\rm on} \ \partial\varOmega\). Some uniform estimates on rates of convergence in C([0,T],L2(Ω)) and C([0,T],H1(Ω)) of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition are obtained.

We consider the problem of a rigid body immersed in an inviscid incompressible fluid in two dimensional space. The motion of the fluid is described by the incompressible Euler equations and the motion of the rigid body is governed by the balance of linear and angular momentum. A global weak solution is obtained, without any assumption on the weighted norm of the initial vorticity.

This paper concerns the instability and stability of the trivial steady states of the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then nonlinear asymptotic stability holds true, otherwise, there is a sharp critical viscosity, which distinguishes the nonlinear stability from instability., 35 pages

We study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see definition in the paper). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy. Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions., Comment: 13 pages, Submitted

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Holder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

In the first part of this paper, we establish the global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen–Frank model. The liquid crystal flow is a prototype of equations from the Ericksen–Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen–Leslie system is a system of the Navier–Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove the global existence of solutions of the Ericksen–Leslie system for a general Oseen–Frank model in R 2 .

In this paper, we study a class of analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients, which describe compressible fluids moving into outer vacuum. For suitable viscous polytropic fluids, we construct a class of radial symmetric and self-similar analytical solutions in R N ( N ⩾ 2 ) with both continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum. Such solutions exhibit interesting new information such as the formation of vacuum at the center of the symmetry as time tends to infinity and explicit regularities and large time decay estimates of the velocity field.

In this paper, we consider the Boltzmann equation with soft potentials and prove the stability of a class of non-trivial profiles defined as some given local Maxwellians. The method consists of the analytic techniques for viscous conservation laws, properties of Burnett functions and the energy method through the micro‐macro decomposition of the Boltzmann equation. In particular, one of the key observations is a detailed analysis of the Burnett functions so that the energy estimates can be obtained in a clear way. As an application of the main results in this paper, we prove the large time nonlinear asymptotic stability of rarefaction waves to the Boltzmann equation with soft potentials.

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum, and are the first for global classical solutions which may have large oscillations and can contain vacuum states., 30 pages

In this paper, we study the linear operator associated with the fluid–rigid body problem. The operator was first introduced by T. Takahashi and M. Tucsnak (2004) [22] . For the general three-dimensional case, we prove that the corresponding semigroup is analytic on L 6 5 ( R 3 ) ∩ L p ( R 3 ) ( p ⩾ 2 ) . In particular, when the solid is a ball in R 3 , the corresponding semigroup is analytic on L 2 ( R 3 ) ∩ L p ( R 3 ) ( p ⩾ 6 ) . And for this case, a unique local strong solution to the fluid–rigid body problem is derived.

The compressible Navier-Stokes system (CNS) with density-dependent viscosity coefficients is considered in multi-dimension, the prototype of the system is the viscous Saint-Venat model for the motion of shallow water. A spherically symmetric weak solution to the free boundary value problem for CNS with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path, which is continuous away from the symmetry center. Detailed regularity and Lagrangian structure of this solution have been obtained. In particular, it is shown that the particle path is uniquely defined starting from any non-vacuum region away from the symmetry center, along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved. Starting from any non-vacuum point at a later-on time, a particle path is also uniquely defined backward in time, which either reaches at some initial non-vacuum point, or stops at a small middle time and connects continuously with vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time, and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up. This finally leads to the formation of vacuum state almost everywhere as the time goes to infinity.

The authors study vanishing viscosity limits of solutions to the 3-dimensional incompressible Navier-Stokes system in general smooth domains with curved boundaries for a class of slip boundary conditions. In contrast to the case of flat boundaries, where the uniform convergence in super-norm can be obtained, the asymptotic behavior of viscous solutions for small viscosity depends on the curvature of the boundary in general. It is shown, in particular, that the viscous solution converges to that of the ideal Euler equations in C([0, T];H1(Ω)) provided that the initial vorticity vanishes on the boundary of the domain.

In this paper, we study the global subsonic irrotational flows in a multi-dimensional ($n\geq 2$) infinitely long nozzle with variable cross sections. The flow is described by the inviscid potential equation, which is a second order quasilinear elliptic equation when the flow is subsonic. First, we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension for sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow. Furthermore, we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely, provided that the incoming mass flux is less than the critical value. This gives a positive answer to the problem of Bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension. Finally, under suitable asymptotic assumptions of the nozzle, we obtain the asymptotic behavior of the subsonic flow in far fields by a blow-up argument. The main ingredients of our analysis are methods of calculus of variations, the Moser iteration techniques for the potential equation and a blow-up argument for infinitely long nozzles., Comment: to appear in Arch. Rational Mech. Anal

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum state. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations., Comment: 30 pages

We extend the well-known Serrin's blowup criterion for the three-dimensional (3D) incompressible Navier–Stokes equations to the 3D viscous compressible cases. It is shown that for the Cauchy problem of the 3D compressible Navier–Stokes equations in the whole space, the strong or smooth solution exists globally if the velocity satisfies the Serrin's condition and either the supernorm of the density or the $L^1(0,T;L^\infty)$-norm of the divergence of the velocity is bounded. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, the Serrin's condition on the velocity can be removed in this criterion.

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Holder estimates for weak solutions by the estimates in the Campanato space. Based on such Holder estimates, we then investigate subsonic-sonic flows with singularities at the sonic curves in a symmetric convergent nozzle with straight wall for an approximate model of the potential flow equation. It is proved that the perturbation problem of the symmetric subsonic-sonic flow is solvable and the symmetric subsonic-sonic flow is stable.

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases.

We establish the existence of a multidimensional transonic shock solution in a class of slowly varying nozzles for the three dimensional steady full Euler system with axially symmetric exit pressure in the diverging part lying in an appropriate scope. We also show that the shock position depends monotonically on the exit pressure.

The large time asymptotic behavior towards viscous contact waves for a class of systems of viscous conservation laws is studied in this paper for general initial perturbations. The high order devia...

In this paper, we obtain a blow up criterion for classical solutions to the 3-D compressible Naiver-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, initial vacuum is allowed in our case., Comment: 25 pages

In this paper, we establish the existence and stability of a 3-D transonic shock solution to the full steady compressible Euler system in a class of de Laval nozzles with a conic divergent part when a given variable axi-symmetric exit pressure lies in a suitable scope. Thus, for this class of nozzles, we have solved such a transonic shock problem in the axi-symmetric case described by Courant and Friedrichs (1948) in Section 147 of [8] : Given the appropriately large exit pressure p e ( x ) , if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e ( x ) .

This work investigates the solvability, regularity and vanishing viscosity limit of the 3D viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition.

In this paper, solutions with nonvanishing vorticity are established for the three dimensional stationary incompressible Euler equations on simply connected bounded three dimensional domains with smooth boundary. A class of additional boundary conditions for the vorticities are identified so that the solution is unique and stable.

In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P r , if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the strength of the shock front are automatically adjusted so that the end pressure at exit becomes P r , as clearly stated by Courant and Friedrichs [Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic in the downstream region Ω+ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 × 3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u2 with a value given at a point, and an algebraic equation on (ρ, u1, u2) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution for a class of nozzles with some large pressure given at the exit.

In this paper, we establish the existence and uniqueness of a transonic shock solution to the full steady compressible Euler system in a class of de Laval nozzles with a large straight divergent part when a given variable exit pressure lies in a suitable range. Thus, for this class of nozzles, we have solved the transonic shock problem posed by Courant-Friedrichs in Section 147 of (5). By introducing a new elaborate iteration scheme, we are able to solve this boundary value problem for a coupled elliptic-hyperbolic system with a free boundary without some stringent requirements in the previous studies. One of the key ingredients in this approach is to solve a boundary value problem for a first order linear system with nonlocal terms and a free parameter. In this paper, we focus on the existence and uniqueness of a transonic shock solu- tion in a de Laval nozzle with a large straight diverging part for the two dimensional full steady compressible Euler system. This is motivated by the following well-known transonic shock phenomena described by Courant-Friedrichs in Section 147 of (5): Given the appropriately large exit pressure pe(x), if the upstream flow is still super- sonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to sub- sonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure becomes pe(x). The 2-D full steady Euler system is

In this paper, we prove that a class of C1-smooth approximate solutions {ue, pe} to the 3D steady axisymmetric Euler equations will converge strongly to 0 in \({L^2_{loc}(R^3)}\) . The main assumptions are that the approximate solutions have uniformly finite energy and approach a constant state at far fields. We also show a Liouville type theorem that there are no non-trivial C1-smooth exact solutions with finite energy and uniform constant state at far fields.

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the Navier�Stokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the Navier�Stokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate , it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of . Thus, this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the L8 norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.

In this paper, we study a transonic shock problem for the Euler flows through a class of 2-D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging (or converging) part. If the supersonic incoming flow is symmetric near the divergent (or convergent) part of the nozzle, then, as indicated in Section 147 of [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ., New York, 1948], there exist two constant pressures P 1 and P 2 with P 1 P 2 such that for given constant exit pressure P e ∈ ( P 1 , P 2 ) , a symmetric transonic shock exists uniquely in the nozzle, and the position and the strength of the shock are completely determined by P e . Moreover, it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting. Furthermore, we establish the global existence, stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure P e when the initial unsteady shock lies in the symmetric diverging part of the 2-D or 3-D nozzle. On the other hand, it is shown that an unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle.

We prove the existence of global weak solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions. We focus on the case where those coefficients vanish on vacuum. The solutions are obtained as limits of solutions in annular regions between two balls, and the equations hold in the sense of distribution in the entire space-time domain. In particular, we prove the existence of spherically symmetric solutions to the Saint–Venant model for shallow water.

This paper concerns with Cauchy problems for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficients. Two cases are considered here, first, the initial density is assumed to be integrable on the whole real line. Second, the deviation of the initial density from a positive constant density is integrable on the whole real line. It is proved that for both cases, weak solutions for the Cauchy problem exist globally in time and the large time asymptotic behavior of such weak solutions are studied. In particular, for the second case, the phenomena of vanishing of vacuum and blow-up of the solutions are presented, and it is also shown that after the vanishing of vacuum states, the globally weak solution becomes a unique strong one. The initial vacuum is permitted and the results apply to the one-dimensional Saint-Venant model for shallow water.

In this paper, we study global subsonic and subsonicsonic flows through a general infinitely long nozzle. First, it is proved that there exists a critical value for the incoming mass flux so that a global uniformly subsonic flow exists in the nozzle as long as the incoming mass flux is less than the critical value. More importantly, we establish some uniform estimates for the deflection angles and the minimum speed of the subsonic flows by combining hodograph transformation and the comparison principle for elliptic equations. With the help of these properties and a compensated compactness framework, we get the existence of a global subsonic-sonic flow solution in the case of the critical incoming mass flux.

where and below ∇· and ∇× denote the div and curl operators respectively, n is the outward normal, and τ is the unit tangential vector of ∂Ω. The investigation of vanishing viscosity limit of solutions of the Navier-Stokes equations both in the two and three spacial dimensional cases is a classical issue. There are two related questions arising from here: one is how to describe the inviscid limiting behavior of the Navier-Stokes equation; and the other is that does the Euler equation can be approximated by the Navier-Stokes equations. In the case that the solution to the ∗This research is supported in part by Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research Grants CUHK-4028/04P, CUHK-4040/02P and CUHK-4279/00P.

The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent $\gamma$ lies in the stability regime $(4/3, 2)$, by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. The results obtained in this paper extend those in \cite{LXZ} of the authors for the constant viscosities to the case of density dependent viscosities which are degenerate at vacuum states., Comment: arXiv admin note: text overlap with arXiv:1506.03906

In this paper, we investigate the uniform regularity of solutions to the 3-dimensional isentropic compressible Navier-Stokes system with free surfaces and study the corresponding asymptotic limits of such solutions to that of the compressible Euler system for vanishing viscosity and surface tension. It is shown that there exists an unique strong solution to the free boundary problem for the compressible Navier-Stokes system in a finite time interval which is independent of the viscosity and the surface tension. The solution is uniform bounded both in $W^{1,\infty}$ and a conormal Sobolev space. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. Based on such uniform estimates, the asymptotic limits to the free boundary problem for the ideal compressible Euler system with or without surface tension as both the viscosity and the surface tension tend to zero, are established by a strong convergence argument., 66pages. arXiv admin note: text overlap with arXiv:1202.0657 by other authors

We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as rotations, for general piecewise Lipschitz initial densities and some suitably constructed initial momentum, we obtain infinitely many global admissible weak solutions. Furthermore, we construct a class of finite-states admissible weak solutions to the Euler system with anti-symmetric sources. Under the additional smallness assumption on the initial densities, we also obtain multiple global-in-time admissible weak solutions for more general sources including damping. The basic framework are based on the convex integration method developed by De Lellis and Szekelyhidi [13] , [14] for the Euler system. One of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term.

Consider the dynamics of a layer of viscous incompressible fluid under the influence of gravity. The upper boundary is a free boundary with the effect of surface tension taken into account, and the lower boundary is a fixed boundary on which the Navier-slip condition is imposed. It is proved that there is a uniform time interval on which the estimates independent of both viscosity and surface tension coefficients of the solution can be established. This then allows one to justify the vanishing viscosity and surface tension limits by the strong compactness argument. In the presence of surface tension, the main difficulty lies in the less regularity of the highest temporal derivative of the mean curvature of the free surface and the pressure. It seems hard to overcome this difficulty by using the vorticity in viscous boundary layers. One of the key observations here is to find that there is a crucial cancelation between the mean curvature and the pressure by using the dynamic boundary condition., Comment: 58 pages. Some typos were corrected and some adjustments were made. arXiv admin note: text overlap with arXiv:1202.0657 by other authors

We study the properties of vacuum states in weak solutions to the compressible Navier–Stokes system with spherical symmetry. It is shown that vacuum states cannot develop later on in time in a region far away from the center of symmetry, provided there is no vacuum state initially and two initially non-interacting vacuum regions never meet each other in the future. Furthermore, a sufficient condition on the regularity of the velocity excluding the formation of vacuum states is given.

We consider the 3-D axisymmetric incompressible Euler equations without swirls with vortex-sheets initial data. It is proved that the approximate solutions, generated by smoothing the initial data, converge strongly in L 2 [ 0 , T ] ; L loc 2 ( R 3 ) provided that they have strong convergence in the region away from the symmetry axis. This implies that if there would appear singularity or energy lost in the process of limit for the approximate solutions, it then must happen in the region away from the symmetry axis. There is no restriction on the signs of initial vorticity here. In order to exclude the possible concentrations on the symmetry axis, we use the special structure of the equations for axisymmetric flows and careful choice of test functions.