We provide a Serrin type blow-up criterion for the 3-D viscous compressible flows with large external potential force. For the Cauchy problem of the 3-D compressible Navier-Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup-norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.
We investigate the 2D combustion model with Dirichlet boundary conditions and slip boundary conditions in bounded domains. The global existence of weak and strong solutions and the uniqueness of strong solutions are obtained provided the initial density is small in some precise sense. Using the energy method and the estimates of boundary integrals, we obtain the a priori bounds if the density and velocity field. In addition, we prove the local existence of the strong solutions via iterative method and the contraction mapping theorem. Finally, we extend the well known Serrion's blowup criterion…
This paper proves a Serrin's type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density and velocity field satisfy some Serrin's type condition, then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally.
This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies the Serrin's condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or…
The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for a generalized resolvent problem on the half-space arising from a compressible fluid model of Korteweg type. Such a compressible fluid model was derived by Dunn and Serrin (1985) and studied by Kotschote (2008) as an initial-boundary value problem.
The aim of this paper is to prove the existence of ${\mathcal R}$-bounded solution operator families for a resolvent problem on the upper half-space arising from a compressible fluid model of Korteweg type with free boundary condition. Such a compressible fluid model was derived by Dunn and Serrin (1985) and studied by Kotschote (2008) as a boundary value problem with non-slip boundary condition.
We consider the existence of strong solution to liquid crystals system in critical Besov space,then give a criterion which is similar to Serrin's criterion on regularity of weak solution to Navier-Stokes equations.
We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.
In this paper, we prove that suitable weak solution $(u,b)$ of the 3-D MHD equations can be extended beyond $T$ if $u\in L^\infty(0,T;L^3(\R^3))$ and the horizontal components $b_h$ of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition.
This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) \cite{3DS}, which can be used as a phase transition model. We distinguish two cases when the dimension N=2 and N=1, in the first case we need that $\frac{1}ρ\in L^{\infty}$, when N=1 we get a weak solution in finite time in the energy space.
In this paper, we establish the existence and uniqueness of local strong solutions to the kinetic Cucker--Smale model coupled with the isentropic compressible Navier--Stokes equation in the whole space. Moreover, the blowup mechanism for strong solutions to the coupled system is also investigated.
We establish a blowup criterion for the two-dimensional (2D) full compressible Navier-Stokes system. The criterion is given in terms of the divergence of the velocity field only, and is independent of the temperature. It is almost the same as that of barotropic case.
In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contain vacuum in any finite regions. Moreover, a simplifie…
In this paper, we shall prove a blow-up criteria in terms of the upper bound of the density, deformation tensor and the gradient of deformation tensor for 3D compressible visco-elasticity. Based on the structure of deformation tensor, we find a cancelation in higher order derivatives, which brings us the desired results.
We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $ρ$ and the divergence of the velocity $\bu$ without any restriction on the physical viscosity constants. Our blow up criteria can be seen as a partial realization of the underlying principle that the higher integrability implies the boundedness and then eventual regularity. We also present similar blow up criterion for the heat conducting gases.
Regularity and uniqueness of weak solutions of the compressible barotropic Navier-Stokes equations with constant viscosity coefficients is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $P(ρ)=aρ^γ$ with $γ>1$ (in dimension three, additional conditions of size will be ask on $γ$). The second part of the paper is devoted to blow-up criteria for slightly subcritical initial data for the scaling of the equations whe…
This paper establishes a blow-up criterion of strong solutions to the two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion depends on the density, but is independent of the velocity and the magnetic field. More precisely, once the strong solutions blow up, the $L^{\infty}$-norm for the density tends to infinity. In particular, the vacuum in the solutions is allowed.
It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a generalization of Xin's work \cite{Xin98} to the isothermal case.
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 h…
We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier-Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is \textit{a priori} estimate for an important quantity under the assumption that the density is upper bounded, whose divergence can be viewed as the effective viscous flux.
We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in $\R^3$. We establish a blow-up criterion for the local strong solutions in terms of the density and magnetic field. Namely, if the density is away from vacuum ($ρ= 0$) and the concentration of mass ($ρ=\infty$) and if the magnetic field is bounded above in terms of $L^\infty$-norm, then a local strong solution can be continued globally in time.
We are concerned with an initial boundary value problem for the compressible magnetohydrodynamic equations with viscosity depending on the density. It is show that for the initial density away from vacuum, the strong solution to the problem exists globally if the gradient of velocity satisfies $\|\nabla\mathbf{u}\|_{L^{2}(0,T;L^\infty)}<\infty$. Our method relies upon the delicate energy estimates and elliptic estimates.
In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of liquid crystal director field.
In this paper, we prove a blow-up criterion in terms of the magnetic field $H$ and the mass density $ρ$ for the strong solutions to the $3$D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the $L^\infty$ norms of $(H,ρ)$ control the possible blow-up (see \cite{olga}\cite{zx}) for strong solutions, which means that if a solution of the compressible isentropic non-resistive MHD equations is initially smooth and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the $L…
In this paper, the 3-D compressible MHD equations with initial vacuum or infinity electric conductivity is considered. We prove that the $L^\infty$ norms of the deformation tensor $D(u)$ and the absolute temperature $θ$ control the possible blow-up (see [5][18][20]) for strong solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $D(u)$ and $θ$ as the critical time approaches. The viscosity coefficients are only restricted by the physic…
We study the low-energy solutions to the 3D compressible Navier-Stokes-Poisson equations. We first obtain the existence of smooth solutions with small $L^2$-norm and essentially bounded densities. No smallness assumption is imposed on the $H^4$-norm of the initial data. Using a compactness argument, we further obtain the existence of weak solutions which may have discontinuities across some hypersurfaces in $\mathbb{R}^3$. We also provide a blow-up criterion of solutions in terms of the $L^\infty$-norm of density.
We consider the non-isentropic compressible Navier-Stokes equations in two or three space dimensions for which the heat conduction of Fourier's law is replaced by Cattaneo's law and the classical Newtonian flow is replaced by a revised Maxwell flow. We show that a physical entropy exists for this new model. For two special cases, we show the global well-posedness of solutions with small initial data and the blow-up of solutions in finite time for a class of large initial data. Moreover, for vanishing relaxation parameters, the solutions (if it exists) are shown to converge to solutions of the …
In this paper, we consider the $3$D compressible radiation hydrodynamic (RHD) equations with thermal conductivity in a bounded domain. The existence of unique local strong solutions is firstly established when the initial data are arbitrarily large and satisfy some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set. Moreover, we show that if the initial vacuum domain is not so irregular, then the compatibility condition is necessary and sufficient to guarantee the existence of the unique strong solution. Finally, …
In the paper, we establish a blow-up criterion in terms of the integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier-Stokes equations in \mathbb{R}^3 with vacuum, under the assumptions on the coefficients of viscosity: \frac{29μ}{3}>λ. This extends the corresponding results in [20, 36] where a blow-up criterion in terms of the upper bound of the density was obtained under the condition 7μ>λ. As a byproduct, the restriction 7μ>λin [12, 37] is relaxed to \frac{29μ}{3}>λfor the full compressible Navier-Stokes equations by giving a new proof of L…
In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$ 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 th…
This paper is to derive a new blow-up criterion for the 2D full compressible Navier-Stokes equations without heat conduction in terms of the density $ρ$ and the pressure $P$. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm $\|ρ||_{{L^\infty(0,t;L^{\infty})}}+||P||_{L^{p_0}(0,t;L^\infty)}<\infty$ for some constant $p_0$ satisfying $1<p_0\leq 2$. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as $\|ρ||_{{L^\infty(0,t;L^{\infty})}}+||P||_{L^{\…
In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier-Stokes equations and the classical Poisson-Nernst-Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover…
This work is devoted to prove existence of global weak solutions for a general isothermal model of capillary fluids derived by J.- E Dunn and J. Serrin (1985) [6], which can be used as a phase transition model. We improve the results of [5] by showing the existence of global weak solution in dimension two for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients. In particular we are interested in capillary coefficients approximating a constant capillarity coefficient. To finish we show the existence of global weak solution in d…
We study the regularity problem of a nematic liquid crystal model with local configuration represented by Q-tensor in three dimensions. It was an open question whether the classical Prodi-Serrin condition implies regularity for this model. Applying a wavenumber splitting method, we show that a solution does not blow-up under certain extended Beale-Kato-Majda condition solely imposed on velocity. This regularity criterion automatically implies that the classical Prodi-Serrin or Beale-Kato-Majda condition prevents blow-up of solutions.
We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations (CNS) under general pressure laws in all dimensions $d\geq 2$. For all hypo-viscosities $(-Δ)^α$ with $α\in (0,1)$, we prove that there exist infinitely many weak solutions with the same initial data. This provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Our proof features new constructions of building blocks for both the density and momentum, which respect the compressible structure. It also applies to the compressible Euler equations and the hypo-viscous i…
In this paper, we prove a Beale--Kato--Majda blow-up criterion in terms of the gradient of the velocity only for the strong solution to the 3-D compressible nematic liquid crystal flows with nonnegative initial densities. More precisely, the strong solution exists globally if the $L^{1}(0,T;L^{\infty})$-norm of the gradient of the velocity $u$ is bounded. Our criterion improves the recent result of X. Liu and L. Liu (\cite{LL}, A blow-up criterion for the compressible liquid crystals system, arXiv:1011.4399v2 [math-ph] 23 Nov. 2010).
The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investiga…
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariants. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial dat…
This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985), which can be used as a phase transition model. We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling spaces, and we prove global existence of solution and uniqueness for data close to a stable equilibrium. For general data, existence and uniqueness is stated on a short time interval. In the general case with physical coefficien…
In this article, we study the blowup phenomena of compressible Euler equations with non-vacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding blowup results of the 1-dimensional case in non-radial symmetry are also included.
We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.
This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous heat conducting MHD equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the absolute temperature and magnetic field. Then as an immediate application, we prove the globa…
We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev-class initial data which are a small $L^ \infty $ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blow up time and location can be explicitly computed, and solutions at the blow up t…
The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global existence and blow-up as well as uniform bounds are reviewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry's version of weighted Gronwall's inequalities, and (vi) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirich…
