Unique Smooth Solution Research Articles

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (∂yyu1,∂xxu2) and the partial magnetic diffusion (∂yyb1,∂xxb2). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique.

Classical solution of the mixed problem for the Klein – Gordon – Fock type equation with characteristic oblique derivatives at boundary conditions

The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with oblique derivatives at boundary conditions in the half-strip is considered. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable func tions were proven for these equations when initial functions are smooth enough. It is proven that fulfilling the matching conditions on the given functions is necessary and sufficient for existence of the unique smooth solution, when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the ori ginal definition area into subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. The obtained solutions are then glued in common points, and the obtained glued сonditions are the matching conditions. Intensification of smoothness requirements for source functions is proven when the di rections of the oblique derivatives at boundary conditions are matched with the directions of the characteristics. This approach can be used in constructing both the analytical solution, when the solution of the integral equation can be found explicitly, and the approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When a numerical solution is constructed, the matching conditions are significant and need to be considered while developing numerical methods.

A non-homogeneous boundary-value problem for the KdV–Burgers equation posed on a finite domain

We consider the initial–boundary value problem for the KdV–Burgers equation posed on a bounded interval [a,b]. This problem features non-homogeneous boundary conditions applied at x=a and x=b and is known to have unique global smooth solution.

Remarks on the three and two and a half dimensional Hall-magnetohydrodynamics system: deterministic and stochastic cases

The Hall-magnetohydrodynamics system has a long history of various applications in physics and engineering, in particular magnetic reconnection, star formation and the study of the sun. Mathematically, the additional Hall term elevates the level of non-linearity to the quasi-linear type while similar systems of equations such as the Navier–Stokes equations, as well as the viscous and magnetically diffusive magnetohydrodynamics system are of the semi-linear type. Consequently, the mathematical analysis on the Hall-magnetohydrodynamics system had been relatively absent for more than half a century, and only started seeing rapid developments in the last few years. In this manuscript, we survey recent progress on the mathematical analysis of the Hall-magnetohydrodynamics system, both in the deterministic and stochastic cases, elaborating on the structure of the Hall term. We list some remaining open problems, and discuss their difficulty. As a byproduct of our discussion, it is deduced that the two and a half dimensional Hall-magnetohydrodynamics system with viscous diffusion in the form of a Laplacian and the magnetic diffusion in the form of a fractional Laplacian with power $$\frac{3}{2}$$ admits a unique smooth solution for all time.

Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. We first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by nonlinear arguments. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic–sonic limit of the solutions is also shown. Finally, through the incompressible limit, we establish the existence and uniqueness of incompressible Euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets. The methods and techniques developed here will be useful for solving other problems involving similar difficulties.

Asymptotic behavior of solutions to bipolar Euler–Poisson equations with time-dependent damping

In this paper, we study the one-dimensional Euler–Poisson equations of bipolar hydrodynamic model for semiconductor device with time-dependent damping effect − J ( 1 + t ) λ for − 1 < λ < 1 , where the damping effect is time-gradually-degenerate for λ > 0 , and time-gradually-enhancing for λ < 0 . Such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time-asymptotic-weakly or strongly. Based on technical observation, and by using the time-weighted energy method, where the weights are artfully chosen, we prove that the system admits a unique global smooth solution, which time-asymptotically converges to the corresponding diffusion wave, when the initial perturbation around the diffusion wave is small enough. The convergence rates are specified in the algebraic forms O ( t − 3 4 ( 1 + λ ) ) and O ( t − ( 1 − λ ) ) according to different values of λ in ( − 1 , 1 7 ) and ( 1 7 , 1 ) , respectively, where λ = 1 7 is the critical point, and the convergence rate at the critical point is O ( t − 6 7 ln ⁡ t ) . All these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are L 2 -integrable. Particularly, when λ = 1 7 , the convergence rate is the fastest, namely, the asymptotic profile of the original system at the critical point is the best.

Classical solution of the mixed problem for the Klein – Gordon – Fock type equation in the half-strip with curve derivatives at boundary conditions

The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with curve derivatives at boundary conditions is considered in the half-strip. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable functions were proven for these equations when initial functions are smooth enough. It is proven that the fulfillment of the matching conditions on the given functions is necessary and sufficient for the existence of the unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the original area of definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the obtained glued conditions are the matching conditions. This approach can be used in constructing as an analytical solution when a solution of the integral equation can be found in an explicit way, so an approximate solution. Moreover, approximate solutions can be constructed in numerical or analytical form. When a numerical solution is built, the matching conditions are essential and they need to be considered while developing numerical methods.

Well-posedness of the free boundary problem for incompressible elastodynamics

The free boundary problem for the three dimensional incompressible elastodynamics system is studied under the Rayleigh–Taylor sign condition. Both the columns of the elastic stress FF⊤−I and the transpose of the deformation gradient F⊤−I are tangential to the boundary which moves with the velocity, and the pressure vanishes outside the flow domain. The linearized equation takes the form of wave equation in terms of the flow map in the Lagrangian coordinate, and the local-in-time existence of a unique smooth solution is proved using a geometric argument in the spirit of [19].

Global existence versus blow‐up results for one dimensional compressible Navier–Stokes equations with Maxwell's law

AbstractWe consider one dimensional isentropic compressible Navier–Stokes equations with constitutive relation of Maxwell's law instead of Newtonion law. For this new model, we show that for small initial data, a unique smooth solution exists globally and converges to the equilibrium state as time goes to infinity. For some large data, in contrast to the situation for classical compressible Navier–Stokes equations, which admits global solutions, we show finite time blow up of solutions for the relaxed system. Moreover, we prove the compatibility of the two systems in the sense that, for vanishing relaxation parameters, the solutions to the relaxed system are shown to converge to the solutions of classical system.

Well-posedness of Hall-magnetohydrodynamics system forced by L $$\acute{\mathrm{e}}$$ e ´ vy noise

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative L $$\acute{\mathrm {e}}$$ vy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability of the global existence of a unique smooth solution may be arbitrarily close to one given the initial data of which its expectation in a certain Sobolev norm is sufficiently small. The proofs go through for the two and a half dimensional case as well. To the best of the authors’ knowledge, an analogous result is absent in the deterministic case due to the lack of viscous diffusion, exhibiting the regularizing property of the noise. Our result may also be considered as a physically meaningful special case of the extension of work of Kim (J Differ Equ 250:1650–1684, 2011) and Mohan and Sritharan (Pure Appl Funct Anal 3:137–178, 2018) from the first-order quasilinear to the second-order quasilinear system because the Hall term elevates the Hall-magnetohydrodynamics system to the quasilinear class, in contrast to the Naiver–Stokes equations that has most often been studied and is semilinear.

Classical solution to the mixed problems for the Klein–Gordon–Fock-type equation with curve derivatives in boundary conditions

The mixed problem for one-dimensional Klein–Gordon–Fock-type equation with curve derivatives in boundary conditions is considered in half-strip. The solution of this problem is reduced to solving the second type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of the twice continuously differentiable functions were proven for these equations when initial functions are smooth enough. It is proven that fulfillment of the matching conditions on the given functions is necessary and sufficient for the existence of the unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis.This method is reduced to the splitting the original area of the definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then obtained solutions are glued in common points, and received glued conditions are the matching conditions. This approach can be used in constructing as analytical solution, in case when solution of the integral equation can be found in explicit way, so for approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When numeric solution is constructed, then matching conditions are essential and they need to be considered while developing numerical methods.

Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity

This paper is concerned with the 3D nonhomogeneous incompressible magneto-micropolar equations with density-dependent viscosity and vacuum in smooth bounded domains. We prove that the system has a unique global smooth solution under the assumption that the initial energy is suitably small.

Global existence of a diffusion limit with damping for the compressible radiative Euler system coupled to an electromagnetic field

We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely the 3D radiative compressible Euler system coupled to an electromagnetic field through the MHD approximation. Assuming the presence of damping together with suitable smallness hypotheses for the data, we prove that this problem admits a unique global smooth solution.

Interior C2 estimates to hessian-type equations on

In this article, we study global C2 estimates to smooth admissible solutions of the Hessian-type equations on and then obtain a unique smooth admissible solution up to a constant. As a special case, the Hessian-type equations contain a Monge-Ampère equation arising in designing a reflecting surface in geometric optics.

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.

From Monge\u2013Amp\xe8re equations to envelopes and geodesic rays in the zero temperature limit

Let (X,theta ) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)-form theta . By a well-known envelope construction this data determines, in the case when the cohomology class [theta ] is pseudoeffective, a canonical theta -psh function u_{theta }. When the class [theta ] is Kähler we introduce a family u_{beta } of regularizations of u_{theta }, parametrized by a large positive number beta , where u_{beta } is defined as the unique smooth solution of a complex Monge–Ampère equation of Aubin–Yau type. It is shown that, as beta rightarrow infty , the functions u_{beta } converge to the envelope u_{theta } uniformly on X in the Hölder space C^{1,alpha }(X) for any alpha in ]0,1[ (which is optimal in terms of Hölder exponents). A generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes. The proofs of the convergence results do not assume any a priori regularity of u_{theta }. Applications to the regularization of omega -psh functions and geodesic rays in the closure of the space of Kähler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of u_{beta } as a “transcendental” Bergman metric.

Smooth solutions to diffusion approximation radiation hydrodynamics equations

In this paper, we consider the smooth solutions to the Cauchy problem for the diffusion approximation radiation hydrodynamics equations in R3. The model describes the interaction of an inviscid gas with photons. The local existence and uniqueness of smooth solutions is obtained by using the energy method with more subtle energy estimates.

Quasi-neutral limit of Euler–Poisson system of compressible fluids coupled to a magnetic field

In this paper, we consider the quasi-neutral limit of a three dimensional Euler-Poisson system of compressible fluids coupled to a magnetic field. We prove that, as Debye length tends to zero, periodic initial-value problems of the model have unique smooth solutions existing in the time interval where the ideal incompressible magnetohydrodynamic equations has smooth solution. Meanwhile, it is proved that smooth solutions converge to solutions of incompressible magnetohydrodynamic equations with a sharp convergence rate in the process of quasi-neutral limit.

Global Regularity and Time Decay for the 2D Magnetohydrodynamic Equations with Fractional Dissipation and Partial Magnetic Diffusion

This paper focuses on a system of the 2D magnetohydrodynamic (MHD) equations with the kinematic dissipation given by the fractional operator $$(-\Delta )^\alpha $$ and the magnetic diffusion by partial Laplacian. We are able to show that this system with any $$\alpha >0$$ always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we make a detailed study on the large-time behavior of these smooth solutions and obtain optimal large-time decay rates. Since the magnetic diffusion is only partial here, some classical tools such as the maximal regularity property for the 2D heat operator can no longer be applied. A key observation on the structure of the MHD equations allows us to get around the difficulties due to the lack of full Laplacian magnetic diffusion. The results presented here are the sharpest on the global regularity problem for the 2D MHD equations with only partial magnetic diffusion.

Global existence and large-time behavior of solutions to the Cauchy problem of one-dimensional viscous radiative and reactive gas

In this paper we consider the global existence and large-time behavior of solutions of one-dimensional viscous radiative and reactive gas with large initial data. In precedent studies the case for the initial-boundary value problem in bounded domain has been investigated and our main purpose focuses on the corresponding problem in unbounded domain. If the initial data is a large perturbation of a non-vacuum constant equilibrium state and is assumed to be without vacuum, mass concentrations, or vanishing temperatures, then we can show that its Cauchy problem admits a unique global smooth non-vacuum solution which tends time-asymptotically to such an equilibrium state. The key point in the analysis is to deduce the uniform positive lower and upper bounds on the specific volume and the absolute temperature.