Unique Local Solution Research Articles

KdV GEOMETRIC FLOWS ON KÄHLER MANIFOLDS

In this paper, we define a kind of KdV (Korteweg–de Vries) geometric flow for maps from a real line ℝ or a circle S1 into a Kähler manifold (N, J, h) with complex structure J and metric h as the generalization of the vortex filament dynamics from a real line or a circle. By Hasimoto transformation, we find that the KdV geometric flow on a Riemann surface of constant Gauss curvature is just classical complex-valued mKdV equation. From the view point of geometric analysis we show that the Cauchy problems of KdV flow on a Kähler manifold admits a unique local solution in suitable Sobolev spaces. In the case the target manifold (N, J, h) with complex structure J and metric h is a certain type of locally Hermitian symmetric space, we show that the KdV flow exists globally by exploiting the conservation laws and semi-conservation law of KdV flow.

Well-posedness for a class of fourth order diffusions for image processing

A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona–Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models.

Well-posedness for the Navier–Stokes–Nernst–Planck–Poisson system in Triebel–Lizorkin space and Besov space with negative indices

This paper is concerned with the well-posedness of the Navier–Stokes–Nerst–Planck–Poisson system (NSNPP). Let sp=−2+n/p. We prove that the NSNPP has a unique local solution (u→,v,w)∈EuT⁎×EvT⁎×EvT⁎ for (u→0,v0,w0) in a subspace, i.e., Vu1×Vv1×Vv1, of F∞−1,2×Bpsp,∞×Bpsp,∞ with ∇⋅u→0=0. We also prove that there exists a unique small global solution (u→,v,w)∈Eu∞×Ev∞×Ev∞ for any small initial data (u→0,v0,w0)∈F˙∞−1,2×B˙psp,∞×B˙psp,∞ with ∇⋅u→0=0.

Solving a non-linear stochastic pseudo-differential equation of Burgers type

In this paper, we study the initial value problem for a class of non-linear stochastic equations of Burgers type of the following form ∂ t u + q ( x , D ) u + ∂ x f ( t , x , u ) = h 1 ( t , x , u ) + h 2 ( t , x , u ) F t , x for u : ( t , x ) ∈ ( 0 , ∞ ) × R ↦ u ( t , x ) ∈ R , where q ( x , D ) is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, f , h 1 , h 2 : [ 0 , ∞ ) × R × R → R are measurable functions, and F t , x stands for a Lévy space-time white noise. We investigate the stochastic equation on the whole space R in the mild formulation and show the existence of a unique local mild solution to the initial value problem by utilising a fixed point argument.

LOCAL WELL-POSEDNESS OF MUSIELA’S SPDE WITH LÉVY NOISE

We determine sufficient conditions on the volatility coefficient of Musiela’s stochastic partial differential equation driven by an infinite dimensional Lévy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight.

Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C ∞ regularity for any positive time.

On the solvability of the compressible Navier–Stokes system in bounded domains

This paper is dedicated to the well-posedness issue for the barotropic Navier–Stokes system with homogeneous Dirichlet boundary conditions in bounded domains of . We aim at considering data in as large a class as possible. Our main result is that if the initial density is bounded away from zero and belongs to some W1,r with r > N, if the initial velocity is in the Besov space (and satisfies a suitable boundary condition), and if the body force is in for some p > 1 then the system has a unique local solution. Our regularity assumptions are consistent with a dimensional analysis which shows that critical data would correspond to r = N and p = 1, and improve an old result by Solonnikov (1980 J. Sov. Math. 14 1120–32).

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $$({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], U(0) = u_0.\end{array}\right.$$ Here, \({(A(t))_{t\in [0,T]}}\) are unbounded operators with domains \({(D(A(t)))_{t\in [0,T]}}\) which may be time dependent. We assume that \({(A(t))_{t\in [0,T]}}\) satisfies the conditions of Acquistapace and Terreni. The functions F and B are nonlinear functions defined on certain interpolation spaces and \({u_0\in E}\) is the initial value. W H is a cylindrical Brownian motion on a separable Hilbert space H. We assume that the Banach space E is a UMD space with type 2. Under locally Lipschitz conditions we show that there exists a unique local mild solution of (SE). If the coefficients also satisfy a linear growth condition, then it is shown that the solution exists globally. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapien, and Zabczyk, to obtain space-time regularity results for the solution U of (SE). For Hilbert spaces E we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second-order stochastic partial differential equation which has been studied by Sanz-Sole and Vuillermot. This leads to several improvements of their result.

Existence and uniqueness of solution in Sobolev space for an unsteady crystal growth problem with zero surface tension

We study an initial value problem for a two-dimensional dendritic crystal growth model with zero surface tension. If the initial data is in Sobolev space H 2 ( R ) , it is proved that an unique local solution exists in proper Sobolev space.

On viscous Burgers-like equations with linearly growing initial data

The Burgers-like equation is considered: \partial_ t u − Delta u + div G ( u ) = 0 in R^n \times (0,T). In this paper we consider the case that the initial data is not bounded at the space infinity. This paper specifies the growth of nonlinear term as G ( r ) ~ r^ 2 for large r . A typical example is the viscous Burgers equation. Our goal is to solve the initial value problem when the initial data may grow linearly at the space infinity. We shall prove that the problem admits a unique local regular solution.

Existence, Asymptotic Behaviour, and Blow up of Solutions for a Class of Nonlinear Wave Equations with Dissipative and Dispersive Terms

Abstract We consider the existence, both locally and globally in time, the asymptotic behaviour, and the blow up of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative and dispersive terms. Under rather mild conditions on the nonlinear term and the initial data we prove that the above-mentioned problem admits a unique local solution, which can be continued to a global solution, and the solution decays exponentially to zero as t →+∞. Finally, under a suitable condition on the nonlinear term, we prove that the local solutions with negative and nonnegative initial energy blow up in finite time.

Jump type Cahn-Hilliard equations with fractional noises

The authors explore a class of jump type Cahn-Hilliard equations with fractional noises. The jump component is described by a (pure jump) Levy space-time white noise. A fixed point scheme is used to investigate the existence of a unique local mild solution under some appropriate assumptions on coefficients.

Damageable contact between an elastic body and a rigid foundation

In this work, the contact problem between an elastic body and a rigid obstacle is studied, including the development of material damage which results from internal compression or tension. The variational problem is formulated as a first-kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field. The existence of a unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, three two-dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme.

A regularity result for a solid–fluid system associated to the compressible Navier–Stokes equations

In this paper we deal with a fluid-structure interaction problem for a compressible fluid and a rigid structure immersed in a regular bounded domain in dimension 3. The fluid is modelled by the compressible Navier–Stokes system in the barotropic regime with no-slip boundary conditions and the motion of the structure is described by the usual law of balance of linear and angular moment.The main result of the paper states that, for small initial data, we have the existence and uniqueness of global smooth solutions as long as no collisions occur. This result is proved in two steps; first, we prove the existence and uniqueness of local solution and then we establish some a priori estimates independently of time.

On freezing of a finite humid porous medium with a heat flux condition considering the factor for phase conversion

AbstractThis work deals with a theoretical mathematical analysis of freezing (desublimation) of moisture in a finite porous medium with heat‐flux condition in x=0. The position of phase change front at time t, given by x =s (t), divides the porous body into two regions. In the first region there is no moisture movement, and in the other one the process of the coupled heat and moisture flows is described by the well known Luikov's system, considering that the factor for phase conversion is non zero. Equivalence between this problem and a system of Volterra integral equations is found. The existence of a unique local solution in time for this problem is also obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Cauchy Problem for the Kadomtsev-Petviashvili (KPII) Equation in Three Space Dimensions

We prove the existence and uniqueness of local solutions in time for the Kadomtsev-Petviashvili equation (KPII) in ℝ3 for initial data in a suitable anisotropic Sobolev space of functions which have essentially 1+ derivative in the variable x and 0+ derivatives in the variables y and z.

Local Well-Posedness of Musiela's SPDE with Lévy Noise

We determine sufficient conditions on the volatility coefficient of Musiela's stochastic partial differential equation driven by an infinite dimensional Levy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight.

Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors

In this paper we study a free boundary problem modelling the growth of nonnecrotic tumors. The main trait of this free boundary problem is that it is essentially multidimensional, so that its well-posedness is hard to establish by using the usual methods in the classical theory of free boundary problems. In this paper we use the functional analysis method based on the theory of analytic semigroups to prove that this problem has a unique local solution in suitable function spaces. Continuous dependence of the solution on the initial data and regularities of the solution can also be easily obtained by using the argument of this paper.

Quasistatic evolution of damage in an elastic body: numerical analysis and computational experiments

The processes of quasistatic evolution of the mechanical state of an elastic body, and the development of material damage which results from internal compression or tension, are modelled and numerically analyzed. The problem is formulated as an elliptic system for the displacements coupled with a parabolic equation for the damage field. The existence of the unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced. Finally, two two-dimensional numerical simulations are performed to show the accuracy of the scheme and the behaviour of the solution.

On freezing of a finite humid porous medium with a heat flux condition

This paper deals with a theoretical mathematical analysis of freezing (desublimation) of moisture in a finite porous medium with a heat flux condition at x = 0 . An equivalence between this problem and a system of Volterra integral equations is found. The existence of a unique local solution in time for this problem is also obtained.