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In this paper, we prove the global well-posedness of solutions for the Cauchy problem of three-dimensional incompressible Navier-Stokes-Landau-Lifshitz equations under the condition that $\|u_0\|_{H^{\frac12}}+\|\nabla d_0\|_{H^{\frac12+\varepsilon}}$ ($\varepsilon>0)$ is sufficiently small. This result can be seen as an improvement of the previous paper [ 20 ].

We study a porous medium equation with dissipative and hyperdiffusive effects and no sign restrictions on the diffusion coefficient.

A boundary value problem is formulated for a stationary model of mass transfer, which generalizes the Boussinesq approximation in the case when the coefficients in the model equations can depend on the concentration of a substance or on spatial variables. The global existence of a weak solution of this boundary value problem is proved. Some fundamental properties of its solutions are established. In particular, the validity of the maximum principle for the substance’s concentration has been proved. Sufficient conditions on the input data of the boundary value problem under consideration, which ensure the local existence of the strong solution from the space H2, and conditions that ensure the conditional uniqueness of the weak solution with additional property of smoothness for the substance’s concentration are established.

In this paper, we prove that the one- dimensional model of reactive micropolar real gas flow and thermal explosion has a solution locally in time. We first define the notion of a generalized solution for the governing initial-boundary value problem. We prove the claim of local existence by deriving a sequence of approximate problems obtained by Faedo-Galerkin projections. A priori estimates allow us to choose small enough time interval of existence, and show that the sequence of approximations is bounded in certain functional spaces, and therefore has a convergent subsequence. In the end, we show that it is precisely this limit that is the solution to the observed problem.

WOS:000754729000004 In this article, we study a plate equation with frictional damping, nonlinear source and time delay. Firstly, we establish the local existence by using the semigroup theory. Then, under suitable conditions, we prove the nonexistence of global solutions for positive initial energy. Time delays often appear in many practical problems such as thermal, economic phenomena, biological, chemical, physical, electrical engineering systems, mechanical applications and medicine.

In this paper we consider the initial boundary value problem for a viscoelastic wave equation with strong damping and logarithmic nonlinearity of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \int ^{t}_{0} g(t-s) \Delta u(x,s)\,ds - \Delta u_{t} (x,t) = \bigl\vert u(x,t) \bigr\vert ^{p-2} u(x,t) \ln \bigl\vert u(x,t) \bigr\vert $$utt(x,t)−Δu(x,t)+∫0tg(t−s)Δu(x,s)ds−Δut(x,t)=|u(x,t)|p−2u(x,t)ln|u(x,t)| in a bounded domain $\varOmega \subset {\mathbb{R}}^{n} $Ω⊂Rn, where g is a nonincreasing positive function. Firstly, we prove the existence and uniqueness of local weak solutions by using Faedo–Galerkin’s method and contraction mapping principle. Then we establish a finite time blow-up result for the solution with positive initial energy as well as nonpositive initial energy.

We consider the scalar semilinear heat equation u t − Δ u = f ( u ) , where f : [ 0 , ∞ ) → [ 0 , ∞ ) is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in L q ( Ω ) for all non-negative initial data u 0 ∈ L q ( Ω ) , when Ω ⊂ R d is a bounded domain with Dirichlet boundary conditions. For q ∈ ( 1 , ∞ ) this holds if and only if lim sup s → ∞ s − ( 1 + 2 q / d ) f ( s ) ∞ ; and for q = 1 if and only if ∫ 1 ∞ s − ( 1 + 2 / d ) F ( s ) d s ∞ , where F ( s ) = sup 1 ≤ t ≤ s ⁡ f ( t ) / t . This shows for the first time that the model nonlinearity f ( u ) = u 1 + 2 q / d is truly the ‘boundary case’ when q ∈ ( 1 , ∞ ) , but that this is not true for q = 1 . The same characterisations hold for the equation posed on the whole space R d provided that lim sup s → 0 f ( s ) / s ∞ .

CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior This research project has as main objective to generalize and improve recently developed methods to establish existence, uniqueness and blow-up criteria of local solutions in time for the Navier-Stokes equations involving Sobolev-Gevrey and Lei-Lin spaces; as well as assuming the existence of a global solution in time for this same system, present decay rates of these solutions in these spaces.

In this paper, we consider the initial boundary value problem of a coupled viscoelastic Kirchhoff-type equations with degenerate damping: {u(tt) - M(parallel to del u parallel to(2)) Delta u+ integral(t)(0) mu(1)(t - s) Delta(s)ds + |u|k +|v|l|ut|p-1ut=f1(u,v), vtt-M(parallel to del v parallel to 2) Delta u+ integral(t)(0) mu 2(t - s) increment v(s)ds + |v|theta +|u|rho |vt|q-1vt=f2(u,v). Firstly, we prove a local existence theorem by using the Faedo-Galerkin approximations. Then, we study blow up of solutions when initial energy is positive.

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hoelder spaces is proved from the Euler-Lagrangian formulation.

The existence of classical solutions to the Cauchy problem for the Boltzmann equation without angular cutoff has been extensively studied in the framework when the solution has Maxwellian decay in the velocity variable, cf. [6, 8] and the references therein. In this paper, we prove local existence of solutions with polynomial decay in the velocity variable for the Boltzmann equation with soft potential. In the proof, the singular change of variables between post- and pre-collision velocities plays an important role, as well as the regular one introduced in the celebrated cancelation lemma by Alexandre–Desvillettes–Villani–Wennberg, Published: 11 December 2013

In this paper we study the Cauchy problem of the Debye system for initial data in the Lorentz space L n , 1 ( R n ) and the Besov space B ˙ L n , 1 , ∞ − α ( R n ) for 0 α 1 . Some local existence theorems are proved.

In the paper, the solvability of the free boundary problem of magnetohydrodynamics for a viscous incompressible fluid in a simply connected domain is proved. The solution is obtained in the Sobolev–Slobodetskii spaces $$ W_2^{2 + l,1 + l/2},1/2 < l < 1 $$ . Bibliography: 15 titles.

In this paper, we investigate the well-posedness of an initial–boundary value problem for the equations of multidimensional radiation hydrodynamics which are a hyperbolic-Boltzmann coupled system. We obtain the local existence and uniqueness of smooth solutions to this problem by using the energy method.

International audience; We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infonity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cuto and non-cuto collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial innity), and solutions in the whole space having a limit equilibrium state at the spatial innity are included in our category.

We consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping.

Веснік Віцебскага дзяржаўнага ўніверсітэта. - 2015. - № 5. - С. 14-19. - Библиогр.: с. 19 (10 назв. )., Цель статьи – исследование локального существования решений начально-краевой задачи. = The aim of this work is to study the local existence of solutions of the initial-boundary value problem.

In the hyper-viscous Navier–Stokes equations of incompressible flow, the operator A=−Δ is replaced by Aα,a,b≡aAα+bA for real numbers α,a,b with α⩾1 and b⩾0. We treat here the case a>0 and equip A (and hence Aα,a,b) with periodic boundary conditions over a rectangular solid Ω⊂R n . For initial data in L p (Ω) with α⩾n/(2p)+1/2 we establish local existence and uniqueness of strong solutions, generalizing a result of Giga/Miyakawa for α=1 and b=0. Specializing to the case p=2, which holds a particular physical relevance in terms of the total energy of the system, it is somewhat interesting to note that the condition α⩾n/4+1/2 is sufficient also to establish global existence of these unique regular solutions and uniform higher-order bounds. For the borderline case α=n/4+1/2 we generalize standard existing (for n=3) “folklore” results and use energy techniques and Gronwall's inequality to obtain first a time-dependent Hα-bound, and then convert to a time-independent global exponential Hα-bound. This is to be expected, given that uniform bounds already exist for n=2,α=1 ([6, pp. 78–79]), and the folklore bounds already suggest that the α⩾n/4+1/2 cases for n⩾3 should behave as well as the n=2 case. What is slightly less expected is that the n⩾3 cases are easier to prove and give better bounds, e.g. the uniform bound for n⩾3 depends on the square of the data in the exponential rather than the fourth power for n=2. More significantly, for α>n/4+1/2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform L2-estimate and are algebraic in terms of the uniform L2-bounds on the initial and forcing data. The integer powers on the square of the data increase without bound as α↓n/4+1/2, thus “anticipating” the exponential bound in the borderline case α=n/4+1/2. We prove our results for the case a=1 and b=0; the general case with a>0 and b⩾0 can be recovered by using norm-equivalence. We note that the hyperviscous Navier–Stokes equations have both physical and numerical application.

We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posedness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions à la Leray, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. Our results extend Giga-Yoshida (1984) [8] ones to the space dimension two, and improve them in terms of requiring less regularity on the velocity fields.

The theory of nonlinear weakly hyperbolic equations was developed during the last decade in an astonishing way. Today we have a good overview about assumptions which guarantee local well posedness in spaces of smooth functions(C∞, Gevrey). But the situation is completely unclear in the case of Sobolev spaces. Examples from the linear theory show that in opposite to the strictly hyperbolic case we have in general no solutions valued in Sobolev spaces. If so-called Levi conditions are satisfied, then the situation will be better. Using sharp Levi conditions ofC∞-type leads to an interesting effect. The linear weakly hyperbolic Cauchy problem has a Sobolev solution if the data are sufficiently smooth. The loss of derivatives will be determined in essential by special lower order terms. In the present paper we show that it is even possible to prove the existence of Sobolev solutions in the quasilinear case although one has the finite loss of derivatives for the linear case. Some of the tools are a reduction process to problems with special asymptotical behaviour, a Gronwall type lemma for differential inequalities with a singular coefficient, energy estimates and a fixed point argument.

In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.

We study the free boundary evolution between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition. The well-posedness of the full water wave problem was first obtained by Wu \cite{Wu}. The methods introduced in this paper allows us to consider multiple cases: with or without gravity, but also a closed boundary or a periodic boundary with the fluids placed above and below it. It is assumed that the initial interface does not touch itself, being a part of the evolution problem to check that such property prevails for a short time, as well as it does the Rayleigh-Taylor condition, depending conveniently upon the initial data. The addition of the pressure equality to the contour dynamic equations is obtained as a mathematical consequence, and not as a physical assumption, from the mere fact that we are dealing with weak solutions of Euler's equation in the whole space., Comment: 44 pages

International audience; In this paper we consider a multi-dimensional damped semiliear wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. We firstly prove the local existence by using the Faedo-Galerkin approximations combined with a contraction mapping theorem. Secondly, the exponential growth of the energy and the $L^p$ norm of the solution is presented.

The Hele-Shaw flows with nonlinear kinetic undercooling regularization were discussed. The velocity vector field v ≡ (vx, vy)of a two-dimensional flow driven by a single sink at the origin with continuous intensity q0 = q0(t) was found. The nonlinear system of equations for the coefficients h̃n was reported.

The existence of time local $C^{\infty}$ -solutions is shown for Cauchy problem of the porous medium equations. Our arguments rely on the " $L^{\infty}$ -energy method" developed in our previous paper [16] and a new method based on the theory of evolution equations in the $L^{2}$ -framework which enables us to handle with perturbations which can be decomposed into monotone parts and small parts in Sobolev spaces of higher order.

In this paper we consider a semilinear hyperbolic equation with source and damping terms. We will prove a blow up result of solutions for positive initial energy.