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In this paper, we consider a Kirchhoff-type parabolic equation with logarithmic nonlinearity. By making a more general assumption about the Kirchhoff function, we establish a new finite time blow-up criterion. In particular, the blow-up rate and the upper and lower bounds of the blow-up time are also derived. These results generalize some recent ones in which the blow-up results were obtained when the Kirchhoff function was assumed to be a very special form.

The Landau-Lifshitz-Bloch-Maxwell equations with polarization describe the evolution of the mean fields in continuous ferromagnetics. In this paper, we firstly use the energy method to prove the existence of weak solution to the Landau-Lifshitz-Bloch-Maxwell equations with polarization for the viscosity problem in two dimensions. Then we prove that the estimates are uniformed in ϵ for solutions to viscosity problem, and letting ϵ → 0 we obtain the global weak solution for the Landau-Lifshitz-Bloch-Maxwell equations with polarization. Finally combining the a p r i o r i estimates, we obtain the existence of smooth solutions for the Landau-Lifshitz-Bloch-Maxwell equations with polarization in two dimensions.

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of an extended modified KdV equation with decaying initial data in two transition regions, completing previous results by Liu et al. in [18] . It turns out that in first region the asymptotics is expressed in terms of second Painleve transcendents, in second region the asymptotics can be given by the sum of the cosine oscillation function and the solution of a Painleve II equation.

Considered in this paper is the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. The main investigation is the Cauchy problem of the new shallow-water model with qualitative properties of its solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model, and then derive a precise blow-up scenario and the blow-up mechanisms for solutions with certain initial profiles. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we investigate the persistence properties of the solution to the Cauchy problem of the new shallow-water model.

In this paper, the existence of the time-periodic solution to the time-dependent Ginzburg–Landau(TDGL) equations in the case of BCS–BEC crossover is established relying on the Leray–Schauder fixed ...

This paper is devoted to studying the global attractors of the periodic initial value problem for Landau–Lifshitz–Bloch–Maxwell system. First we give the global existence of the smooth solution for this system. Then, we prove the existence of global attractors, the Hausdorff dimension and fractal dimension have been estimated.

The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou.

In this paper, by adapting the method introduced in H. Kozono and S. Shimizu (Strong solutions of the Navier–Stokes equations based on the maximal Lorentz regularity theorem in Besov spaces. J Func...

In this paper, we are concerned with an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force. We first establish the local well-posedness in a range of the Besov spaces, as well as the local well-posedness in the critical space. Then, we study the Gevrey regularity of the shallow-water model by using a generalized Cauchy-Kovalevsky theorem, which implies that the shallow-water model admits analytical solutions locally in time and globally in space. Moreover, we obtain a precise lower bound of the lifespan and the continuity of the solution. Finally, working with moderate weight functions that are commonly used in time-frequency analysis, some persistence results to the shallow-water model are illustrated.

We consider the dynamics of an Boussinesq approximation Benard convection fluid evolving in a three-dimensional domain bounded below by a fixed flatten boundary and above by a free moving surface. The domain is horizontally periodic and the effect of the surface tension is on the free surface. By developing a priori estimates for the model, we prove the exponential decay of solutions in the framework of high regularity.

This paper deals with the Cauchy problem for the generalized Benjamin–Ono–Burgers equation ∂tu+H∂x2u−νuxx+∂x(uk+1/(k+1))=0, k⩾ 4, where H denotes Hilbert transform. We obtain its global well-posedn...

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation ((alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Holder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored., Comment: 54 pages, 0 figure, J. Dyn. Differ. Equ (to appear, 2021)

We consider the long-time behavior for stochastic 2D nematic liquid crystals flows with the velocity field perturbed by an additive noise. The presence of the noises destroys the basic balance law of the nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow even in the weak solution space under non-periodic boundary conditions. To overcome the difficulty we use a new technique some kind of logarithmic energy estimates to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. Considering the existence of random attractor, the common method is to derive uniform a priori estimates in functional space which is more regular than the solution space. We can follow the common method to prove the existence of random attractor in the weak solution space. However, if we consider the existence of random attractor in the strong solution space, it is very difficult and very complicated for such highly non-linear stochastic system with no basic balance law and non-periodic boundary conditions. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to obtain the existence of the random attractor in the strong solution space, which implies the support of the invariant measure lies in a more regular space. As far as we know, it is the first article to attack the long-time behavior of stochastic nematic liquid crystals.

The Cauchy problem for the 3D compressible Navier–Stokes and magnetohydrodynamic equations without heat conductivity is considered. Existence of global-in-time smooth solutions is established under...

Applying similar arguments as introduced in Bona–Smith [4] , we mainly study the inviscid limit of the inhomogeneous incompressible Navier–Stokes equations in R d . Some results of local uniform estimates on solutions of the Navier–Stokes equations, independent of the viscosity, are also obtained.

In this paper, we consider the incompressible limit of strong solution for the compressible nematic liquid crystal flows in a 3D bounded domain Ω⊂R3 where the the velocity field satisfies the Diric...

We consider the Hirota equation on the quarter plane with the initial and boundary values belonging to the Schwartz space. The goal of this paper is to study the long-time behavior of the solution of this initial–boundary value problem based on the asymptotic analysis of an associated matrix Riemann–Hilbert problem.

We develop a Riemann–Hilbert approach to the Cauchy problem on the line for a new type of coupled nonlinear Schrodinger (CNLS) equations i q 1 , t + q 1 , x x + 2 ( | q 1 | 2 − 2 | q 2 | 2 ) q 1 − 2 q 2 2 q 1 ⁎ = 0 , i q 2 , t + q 2 , x x + 2 ( 2 | q 1 | 2 − | q 2 | 2 ) q 2 + 2 q 1 2 q 2 ⁎ = 0 . This approach allows us to give a representation of the solution to the Cauchy problem of the CNLS equations in terms of the solution of a 4 × 4 Riemann–Hilbert problem formulated in the complex k-plane. Due to the energy conservation law of above system is ∫ − ∞ + ∞ ( | q 1 | 2 − | q 2 | 2 ) d x , it is difficult to obtain a solution for this system by using the energy estimate method of PDE's. Therefore, this approach efficiently provides a new way in studying the nonlinear problems that PDE's theory can't solve. Furthermore, this representation is then used for retrieving the soliton solutions.

Under investigation in this paper is a fourth-order dispersive nonlinear Schrodinger equation, which has the physical descriptions in Heisenberg ferromagnetic spin. We first establish the associated Riemann-Hilbert problem on the line with the initial value locating in the Schwartz space. Through the nonlinear steepest descent analysis, the long-time asymptotic behaviors of the solution for this related Cauchy problem are discussed.

In this paper we study the regularity properties of Hirota equation on the right half line with data of low regularity. In particular, using an explicit solution formula of the initial and boundary value problem and the restricted norm method, we prove the local existence, uniqueness, and continuous dependence on initial data in X s , b spaces. Moreover, we obtain the global existence and that the nonlinearity of Hirota equation on the half line is smoother than the initial data.

In this paper, we consider the existence of the suitable weak solutions for the Cauchy problem of the double-diffusive convection system in R3 . We establish the global existence of the suitable we...

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well posed in Sobolev...

We present a Riemann–Hilbert approach for the modified short pulse equation qxt=q+12q(q2)xx on the line. This approach allows us to give a representation of the solution to the Cauchy problem, whic...

In this paper we establish the existence of vortex solutions for a relativistic self-dual U ( 1 ) × U ( 1 ) Chern–Simons system over a doubly periodic domain. We use a constrained minimization approach to obtain the existence result, the system admits a solution when the Chern–Simons coupling parameter κ > 0 is sufficiently small; while no solution exists when κ > 0 is not sufficiently small. And we also get the quantized integral.

We consider the Cauchy problem of the ideal MHD system with a constant non-zero background magnetic field e . We provide a new proof of global existence for small perturbations of an initial stable state ( 0 , e ) . The proof is based on the null structure of the system and some new weighted energy estimates established along the characteristics.

We investigate the nonlinear thermal instability of the magnetohydrodynamic problem for a full compressible viscous fluid with zero resistivity and zero heat conductivity in the presence of a uniform gravitational force in a bounded domain Ω ∈ R 3 . We establish that under some instability conditions, the equilibrium-state is linearly unstable by constructing a suitable energy functional and employing the modified variational method. Then, on the basis of the constructed linearly unstable solutions and the local well-posedness of classical solutions to the original nonlinear problem, we reconstruct the initial data of linearly unstable solutions to be the one of the original nonlinear problem and obtain an appropriate energy estimate of Gronwall-type. Finally, we establish that the equilibrium-state is nonlinearly unstable in view of the energy estimate by a bootstrap method.

The Cauchy problem of the fifth order Kadomtsev–Petviashvili-I equation (fifth-KP-I) (0.1) ∂ t u + ∂ x 5 u ± ∂ x 3 u − ∂ x − 1 ∂ y y u + ∂ x ( u 2 ) = 0 , ( x , y , t ) ∈ R 3 ; is considered. It follows that the fifth order Kadomtsev–Petviashvili-I equation (0.1) is locally well-posed in H s , 0 with s ≥ − 3 / 4 , where the norm H s , r is defined by ‖ f ‖ H ( x , y ) s , r = ‖ 〈 ξ 〉 s 〈 ζ 〉 r f ˆ ‖ L ( ξ , ζ ) 2 .

The Cauchy problem for the three-dimensional compressible magneto-micropolar fluid equations is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data are small perturbations of some given constant state. Moreover, we obtain the time decay rates of the higher-order spatial derivatives of the solution by combining the L p - L q estimates for the linearized equations and the Fourier splitting method, if the initial perturbation is small in H 3 -norm and bounded in L 1 -norm.

In this paper, we prove the existence of a sign-changing solution for a p-Laplacian elliptic problem with constraint in R N . By constructing a pseudo gradient vector field and a descending flow, we obtain three solutions and among these solutions, one is positive, one is negative and one is sign-changing.

We consider a two-layer quasi-geostrophic model in geophysical fluid dynamics. By Faedo–Galerkin method and asymptotic argument, we prove the existence of the global solution to the initial value problem of this model in [Formula: see text]. Moreover, using the Fourier splitting method, we also obtain the decay rate of the solutions.

Consider the Cauchy problems for the following two-dimensional nonlinear singular system of differential equations arising from geostrophics ∂∂t[γ(ψ1−ψ2)−△ψ1]+α(−△)ρψ1+β∂ψ1∂x+J(ψ1,γ(ψ1−ψ2)−△ψ1)=0,∂∂t[γδ(ψ2−ψ1)−△ψ2]+α(−△)ρψ2+β∂ψ2∂x+J(ψ2,γδ(ψ2−ψ1)−△ψ2)=0,ψ1(x,y,0)=ψ01(x,y),ψ2(x,y,0)=ψ02(x,y). In this system, α>0, γ>0, δ>0 and ρ>0 are positive constants, β≠0 is a real nonzero constant, the Jacobian determinant is defined by J(p,q)=∂p∂x∂q∂y−∂p∂y∂q∂x. The existence and uniqueness of the global smooth solution of the system of differential equations are very important in applied mathematics and geostrophics, but they have been open for a long time. The singularity generated by the linear parts, the strong couplings of the nonlinear functions and the fractional order of the derivatives make the existence and uniqueness very difficult to study. Very recently, we found that there exist a few special structures in the system. The main purpose of this paper is to couple together the special structures and an unusual method for establishing the uniform energy estimates to overcome the main difficulty to accomplish the existence and uniqueness of the global smooth solution: ψ1∈C∞(R2×R+) and ψ2∈C∞(R2×R+), for all ρ>3/2. The new energy method enables us to make complete use of the special structures of the nonlinear singular system. The results obtained in this paper provide positive solutions to very important open problems and greatly improve many previous results about nonlinear singular systems of differential equations arising from geostrophics.

Under non-periodic boundary conditions, we consider the long-term behavior of stochastic two-dimensional (2D) nematic liquid crystal flows with multiplicative noise. The structure of the corresponding stochastic model is different from that of the deterministic model. Noise destroys the basic balance law for the nematic liquid crystal flows, so we cannot employ the standard argument to obtain uniform a priori estimates of the solutions to this stochastic model. To overcome this problem, we use logarithmic energy estimates and the Ito formula in a Banach space to obtain uniform estimates that improve the previous result where the orientation field grows exponentially w.r.t. time t. In order to study the existence of a random attractor, we need to show that the solution is a stochastic flow. However, this is not obvious because of the particularly multiplicative noise in the orientation field. To show the flow property of the orientation field, we construct linear stochastic partial differential equations where the solutions are stochastic flows. After determining the relationship between these linear equations and the orientation field equation, we prove that each component of the orientation field is indeed a stochastic flow. The global well-posedness of the stochastic 2D nematic liquid crystal flows is only established for weak solutions, so the existence of the random attractor should be demonstrated in the weak solution space. Thus, using the standard method, we need to derive uniform a priori estimates in the strong solution space. However, the standard method fails because of the ill-posedness of the strong solution. We show that by proving the compactness property of the stochastic flow and using the regularity of the solutions, we can construct a compact absorbing ball in the weak solution space, which implies the existence of the random attractor. Finally, we show the existence of the invariant measure for the corresponding stochastic model, which is the first result for the long-term behavior of stochastic nematic liquid crystals under Dirichlet boundary conditions for the velocity field and Neumann boundary conditions for the orientation field.

The Cauchy problem for the three-dimensional non-isothermal model for compressible nematic liquid crystals is considered. Existence of global-in-time smooth solutions is established provided that the initial datum is close to a steady state ( ρ ¯ , 0 , d ¯ , θ ¯ ) . By using the L q – L p estimates and the Fourier splitting method, if the initial perturbation is small in H 3 -norm and bounded in L q ( q ∈ [ 1 , 6 5 ) ) norm, we obtain the optimal decay rates for the first and second order spatial derivatives of solutions. In addition, the third and fourth order spatial derivatives of director field d in L 2 -norm are achieved.

In this paper, we are concerned with \begin{document}$N$\end{document} -Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

In this paper, we investigate the existence of classical solution of the viscous bipolar quantum hydrodynamic(QHD) models for ir-rotational fluid in a periodic domain. By applying the iteration method, we prove that the viscous bipolar QHD model has a local classical solution. Then we prove this solution is global with small initial data, based on a series of a priori estimates. Finally, we obtained the inviscid limit of this viscous quantum hydrodynamic model.

This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.

In this paper we firstly investigate the local-in-time existence of smooth solution for the quantum hydrodynamic models (QHD) in R d . Then we prove that any smooth solution of the QHD model which satisfies suitable conditions will blow up in finite time. The model can be derived from nonlinear Schrodinger equation by a Madelung transformation. The main idea is based on the construction of approximate solution and energy inequality.

This paper deals with the Cauchy problem for the generalized Camassa–Holm equation. The existence of weak solutions for the generalized Camassa–Holm equation in lower order Sobolev space with is obtained.

The present paper is devoted to the study of the global solution and nonlinear stability to the coupled complex Ginzburg–Landau and Burgers (CGL–Burgers) equations for sequential flames which describe the interaction of the excited oscillatory and the damped monotonic mode governing a sequential chemical reaction. If the solution blows up in finite time, we derive the lower bound of blow-up rate of blow-up solution.

This work is devoted to the Cauchy problem for Schrodinger-damped Boussinesq System and its fractional form in one dimension. We obtained local mild solutions of the Systems by means of the contraction mapping principle. Meanwhile, sharp results of the existence and nonexistence of global mild solutions have been presented.