Search

We provide a new ε -condition for the vorticity of a suitable weak solution to the Navier–Stokes equations that leads to partial regularity. This refines the well known limsup condition of the Caffarelli-Kohn-Nirenberg Theorem by a new condition on the vorticity, replacing limsup by a suitable range of the radius r of the parabolic cylinders. As a consequence, the partial regularity is obtained directly from this ε -condition of the vorticity without relying on the ε -condition of the velocity. Furthermore, by the local nature of the method this result holds for any local suitable weak solution of the Navier–Stokes equations in a general domain. [ABSTRACT FROM AUTHOR]

We remove (α , λ) -discretely self-similar blow up for solutions to the Euler equations for α ≥ 3 2 , for which we allow sublinear growth for the profile. More precisely, we show that there are only spatial constant (α , λ) -discretely self-similar solutions v = c (t) having the sublinear growth at the infinity. For the proof, we establish a new a priori L loc 2 (R 3) estimate for the 3D Euler equations. [ABSTRACT FROM AUTHOR]

In this paper we study the Liouville type problem for stationary magnetohydrodynamic equations in R n . We show that under certain anisotropic integrability conditions on the components of the velocity and the magnetic field the solution is trivial. [ABSTRACT FROM AUTHOR]

We prove a localized Beale–Kato–Majda type blow-up criterion for the 3D incompressible Euler equations in the Hölder space setting. More specifically, let v ∈ C ([ 0 , T) ; C 1 , α (Ω)) ∩ L ∞ (0 , T ; L 2 (Ω)) be a solution to the Euler equations in a domain Ω ⊂ R 3 . If there exists a ball B ⊂ Ω such that ∫ 0 T ‖ ω (s) ‖ B M O (B) d s < + ∞ , where ω = ∇ × v stands for the vorticity, then v ∈ C ([ 0 , T ] ; C 1 , α (K)) for every compact subset K ⊂ B . In the proof of this result, in order to handle the time evolution of the local Hölder norm of the vorticity we use the well-known Campanato space representation for the the Hölder space, and our argument relies on the Campanato space estimates for the solution to the corresponding transport equation. [ABSTRACT FROM AUTHOR]

We obtain a new regularity criterion in terms of the oscillation of time derivative of the pressure for the 3D Navier–Stokes equations in a domain D ⊂ R 3 . The key observation for the proof is that the head pressure defined by Q ¯ (x , t) = 1 2 | v (x , t) | 2 + p ¯ (x , t) , where p ¯ (x , t) = p (x , t) - ∫ t 0 t sup y ∈ Ω (∂ s p (y , s) - | ω (y , s) | 2) d s with p(x, t) the pressure and ω the vorticity satisfies parabolic maximum principle in Ω × (t 0 , T) with Ω ⋐ D . [ABSTRACT FROM AUTHOR]

We prove a localized non blow-up theorem of the Beale–Kato–Majda type for the solution of the 3D incompressible Euler equations. [ABSTRACT FROM AUTHOR]

We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions |$u\in C([0, T); W^{2,q} (\mathbb R^3))$|⁠ , |$q>3$| of the incompressible Euler equations. We show that a blow up at |$t=T$| happens only if $$\begin{align*} &\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \exp \left(\int_{s} ^t \int_0 ^{\sigma} \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \textrm{d}\sigma \right) \right\} \textrm{d}s \textrm{d}t \, = +\infty.\end{align*}$$ As consequences of this criterion we show that there is no blow up at |$t=T$| if |$ \|D^2 p(t)\|_{L^\infty } \le \frac{c}{(T-t)^2}$| with |$c<1$| as |$t\nearrow T$|⁠. Under the additional assumption of |$\int _0 ^T \|u(t)\|_{L^\infty (B(x_0, \rho))} \textrm{d}t <+\infty $|⁠ , we obtain localized versions of these results. [ABSTRACT FROM AUTHOR]

We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions |$u\in C([0, T); W^{2,q} (\mathbb R^3))$|⁠ , |$q>3$| of the incompressible Euler equations. We show that a blow up at |$t=T$| happens only if $$\begin{align*} &\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \exp \left(\int_{s} ^t \int_0 ^{\sigma} \|D^2 p (\tau)\|_{L^\infty} \textrm{d}\tau \textrm{d}\sigma \right) \right\} \textrm{d}s \textrm{d}t \, = +\infty.\end{align*}$$ As consequences of this criterion we show that there is no blow up at |$t=T$| if |$ \|D^2 p(t)\|_{L^\infty } \le \frac{c}{(T-t)^2}$| with |$c<1$| as |$t\nearrow T$|⁠. Under the additional assumption of |$\int _0 ^T \|u(t)\|_{L^\infty (B(x_0, \rho))} \textrm{d}t <+\infty $|⁠ , we obtain localized versions of these results. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Liouville-type theorem for the stationary MHD and the stationary Hall-MHD systems. Assuming suitable growth condition at infinity for the mean oscillations for the potential functions, we show that the solutions are trivial. These results generalize the previous results obtained by two of the current authors in Chae and Wolf (J Differ Equ 295:233–248, 2021). To prove our main theorems, we use a refined iteration argument. [ABSTRACT FROM AUTHOR]

In this paper, we derive kinematic relations for quantities involving the rate of strain tensor and the Hessian of the pressure for solutions of the 3D Euler equations and the 2D Boussinesq equations. Using these kinematic relations, we prove new blow-up criteria and obtain conditions for the absence of type I singularity for these equations. We obtain both global and localized versions of the results. Some of the new blow-up criteria and type I conditions improve previous results of Chae and Constantin (Int Math Res Notices rnab014, 2021). [ABSTRACT FROM AUTHOR]

In this paper we prove Liouville type theorem for the stationary Navier–Stokes equations in R 3 under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth solution (u, p) of the stationary Navier–Stokes equations satisfying u (x) → 0 as | x | → + ∞ and the condition of finite Dirichlet integral ∫ R 3 | ∇ u | 2 d x < + ∞ is trivial, if either | u (x) | / | Q (x) | = O (1) or | p (x) | / | Q (x) | = O (1) as | x | → ∞ , where | Q | = 1 2 | u | 2 + p is the head pressure. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R 3 , having the diffusion term A p (u) = ∇ · (| D (u) | p - 2 D (u)) with D (u) = 1 2 (∇ u + (∇ u) ⊤) , 3 / 2 < p < 3 . In the case 3 / 2 < p ≤ 9 / 5 , we show that a suitable weak solution u ∈ W 1 , p (R 3) satisfying lim inf R → ∞ | u B (R) | = 0 is trivial, i.e., u ≡ 0 . On the other hand, for 9 / 5 < p < 3 we prove the following Liouville type theorem: if there exists a matrix valued function V = { V ij } such that ∂ j V ij = u i (summation convention), whose L 3 p 2 p - 3 mean oscillation has the following growth condition at infinity, ∫ - B (r) | V - V B (r) | 3 p 2 p - 3 d x ≤ C r 9 - 4 p 2 p - 3 ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the L 2 norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale. [ABSTRACT FROM AUTHOR]

In this paper we obtain new local blow-up criterion for smooth axisymmetric solutions to the three dimensional incompressible Euler equation. If the vorticity satisfies ∫ 0 t ∗ (t ∗ - t) ‖ ω (t) ‖ L ∞ (B (x ∗ , R 0)) d t < + ∞ for a ball B (x ∗ , R 0) away from the axis of symmetry, then there exists no singularity at t = t ∗ in the torus T (x ∗ , R) generated by rotation of the ball B (x ∗ , R 0) around the axis. This implies that possible singularity at t = t ∗ in the torus T (x ∗ , R) is excluded if the vorticity satisfies the blow-up rate ‖ ω (t) ‖ L ∞ (T (x ∗ , R)) = O 1 (t ∗ - t) γ as t → t ∗ , where γ < 2 , and the torus T (x ∗ , R) does not touch the axis. [ABSTRACT FROM AUTHOR]

We prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations. [ABSTRACT FROM AUTHOR]

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution v∈L∞(-1,0;L2(B(x0,r)))∩Lloc∞(-1,0;W1,∞(B(x0,r))) of the 3D Euler equations, where B(x0,r) is the ball with radius r and the center at x0, if the limiting values of certain scale invariant quantities for a solution v(·, t) as t→0 are small enough, then ∇v(·,t) does not blow-up at t = 0 in B(x0, r). [ABSTRACT FROM AUTHOR]

We study the scenario of discretely self-similar blowup for Navier–Stokes equations. We prove that at the possible blow-up time such solutions have only one point singularity. In case of the scaling parameter λ near 1, we remove the singularity. [ABSTRACT FROM PUBLISHER]

We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $${\mathbb{R}^2}$$ . We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations. [ABSTRACT FROM AUTHOR]

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ru̇žička-Šverák and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $${L^{p, \infty} (\mathbb{R}^3)}$$ with $${p > \frac{3}{2}}$$ . [ABSTRACT FROM AUTHOR]

The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $${(-\Delta)^\alpha}$$ . Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $${\alpha\ge 1}$$ in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $${\alpha > \frac{1}{2}}$$ . The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations. .. [ABSTRACT FROM AUTHOR]

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $${\Omega \subset \mathbb{R}^3}$$ . In particular, we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $${\Omega=\mathbb{R}^3}$$ , we show that the set of possible singularities is compact. [ABSTRACT FROM AUTHOR]

We present a simple, short, and elementary proof that if v is a Beltrami flow with a finite energyin R3,then v=0. In the case of the Beltrami flows satisfying v∈Lloc∞(R3)∩Lq(R3) with q ∈[2,3),or [ABSTRACT FROM AUTHOR]

We prove the global well posedness of the continuously stratified inviscid quasi-geostrophic equations in $$\mathbb {R}^3$$ . [ABSTRACT FROM AUTHOR]

We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ3. The nonlinear term of these new model equations is obtained slightly modifying that of the Navier-Stokes equations. The system has the same scaling property and Galilean invariance as the Navier-Stokes equations. For such a system we prove the global regularity for smooth initial data. [ABSTRACT FROM AUTHOR]

In this paper we use maximum principle in the far field region for the time dependent self-similar Euler equations to exclude discretely self-similar blow-up for the Euler equations of the incompressible fluid flows. Our decay conditions near spatial infinity of the blow-up profile are given explicitly in terms the coefficient in the equations. We also deduce triviality of the discretely self-similar solution to the magnetohydrodynamic system, under suitable decay conditions near spatial infinity than the previous one. Applying similar argument directly to the Euler equations, we obtain a priori estimate of the vorticity in the far field region. [ABSTRACT FROM AUTHOR]

In this note, we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile. [ABSTRACT FROM AUTHOR]

We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations. [ABSTRACT FROM AUTHOR]

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $${\mathbb {R}^N}$$ . If we assume 'single signedness condition' on the force, then we can show that a $${C^1 (\mathbb {R}^N)}$$ solution ( v, p) with $${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$$ is trivial, v = 0. For the solution of the steady Navier-Stokes equations, satisfying $${v(x) \to 0}$$ as $${|x| \to \infty}$$ , the condition $${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$$ , which is stronger than the important D-condition, $${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$$ , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203-215, ), using the self-similar Euler equations directly. [ABSTRACT FROM AUTHOR]

The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L-condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables $${u \in L^p}$$ and $${u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}$$, then the blow-up does not occur, provided $${\alpha > N/2}$$ or $${-1 < \alpha \leq N\,/p}$$. This includes the L case natural for the Navier-Stokes equations. For $${\alpha = N\,/2}$$ we exclude profiles with asymptotic power bounds of the form $${ |y|^{-N-1+\delta} \lesssim |u(y)| \lesssim |y|^{1-\delta}}$$. Solutions homogeneous near infinity are eliminated, as well, except when homogeneity is scaling invariant. [ABSTRACT FROM AUTHOR]

The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included. [ABSTRACT FROM AUTHOR]

In this paper we study the Euler-Poincaré equations in $${\mathbb {R}^N}$$ . We prove local existence of weak solutions in $${W^{2,p}(\mathbb {R}^N), p > N}$$ , and local existence of unique classical solutions in $${H^k (\mathbb {R}^N)}$$ , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation ( α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to $${C([0, T);H^k(\mathbb {R}^N))}$$ with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution $${\mathbf{u} \in H^1(\mathbb{R}^N)}$$ is u=0; for α= 0 any weak solution $${\mathbf{u} \in L^2(\mathbb{R}^N)}$$ is u=0. [ABSTRACT FROM AUTHOR]

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u related to the scalar θ by $u=\nabla^\perp\Lambda^{\beta-2}\theta$, where $1<\beta\le 2$ and $\Lambda=(-\Delta)^{1/2}$ is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch-type solutions. The second family is a dissipative active scalar equation with $u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper we derive various sufficient conditions on the pressure for vanishing velocity in the incompressible Navier-Stokes and the Euler equations in ℝ N . [ABSTRACT FROM PUBLISHER]

Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component u of the velocity field u is determined by the scalar θ through $${u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}$$ , where $${\mathcal{R}}$$ is a Riesz transform and Λ = (−Δ). The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound $${\|\nabla u||_{L^\infty}}$$ for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log( I + log( I − Δ))) with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ with 0 ≦ β ≦ 1 is also obtained. [ABSTRACT FROM AUTHOR]

We generalize and localize the previous results by the author on the study of self-similar singularities for the 3D Euler equations. More specifically we extend the restriction theorem for the representation for the vorticity of the Euler equations in a bounded domain, and localize the results on asymptotically self-similar singularities. We also present progress towards relaxation of the decay condition near infinity for the vorticity of the blow-up profile to exclude self-similar blow-ups. The case of the generalized Navier-Stokes equations having the laplacian with fractional powers is also studied. We apply the similar arguments to the other incompressible flows, e.g. the surface quasi-geostrophic equations and the 2D Boussinesq system both in the inviscid and supercritical viscous cases. [ABSTRACT FROM AUTHOR]

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations if the vorticity decays sufficiently fast near infinity in $$\mathbb{R}^3$$ . By a similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $$\mathbb{R}^n$$ . This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions. [ABSTRACT FROM AUTHOR]

We study SU(3) Toda system in non-abelian relativistic self-dual gauge theory. In the range of parameters where the corresponding Trudinger-Moser inequality fails, we show the existence of the solution by a different variational formulation from Lucia-Nolasco's [15]. [ABSTRACT FROM AUTHOR]

We prove the local in time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler equations of inviscid incompressible fluid flows in Rn, n≥2. As a corollary we obtain the persistence of Besov space regularity for the solutions of the 2‐D Euler equations with initial velocity belonging to the Besov spaces. For the proof of the results we establish a logarithmic inequality of the Beale–Kato–Majda type and a Moser type of inequality in the Besov spaces. [ABSTRACT FROM AUTHOR]

We consider the Euler system for inviscid incompressible fluid flows, and its perturbations in ℝn, n≥2. We prove global well-posedness of this perturbed Euler system in the Triebel-Lizorkin spaces for initial vorticity which is small in the critical Besov norms. Comparison type theorems about the blow-up of solutions are proved between the Euler system and its perturbations. We also study the possiblity of the self-similar type of blow-up of solutions to the equations. [ABSTRACT FROM AUTHOR]

In this article we prove the global existence of solution of the classical MaxwellChern-Simons-Higgs equations in (2 + 1 )-dimensional Minkowski space-time in the Lorentz gauge. We also prove that the topological solution of the MaxwellChern-Simons-Higgs system converges to that of Maxwell-Higgs system as the Chern-Simons constant k goes to zero, reproducing the classical result by Moncrief [J. Math. Phys. 21, 2291 (1980)] on the global existence of the MaxwellKlein-Gordon system in (2 + 1 )-dimension. [ABSTRACT FROM AUTHOR]

We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

We construct a general type of multivortex solutions of the self-duality equations (the Bogomol'nyi equations) of (2+1) dimensional relativistic Chern–Simons model with the non-topological boundary condition near infinity. For such construction we use a perturbation argument around the explicit solutions of the Liouville equation. [ABSTRACT FROM AUTHOR]

In this Letter, we investigate multiple existence of the multivortex solutions of the self-dual Chern–Simons CP(1) model on a flat torus for each range of the symmetry breaking parameter. We also study asymptotics of solutions as the Chern–Simons coupling constant goes to zero. [ABSTRACT FROM AUTHOR]

We consider the notion of a functional solution of the Euler equations for incompressible fluid flows. We show that a functional solution can be constructed under 'very weak' a priori estimates on approximate solution sequences of the equation; an estimate uniform in L together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in L available for the approximate solution sequence, then the structured functional solution just described becomes a measure-valued solution in the sense of DiPerna & Majda. We also show that a functional solution can be obtained from a measure-valued solution. We give an example showing that a much higher concentration of energy than in the case of measure-valued solutions is allowed by the approximation procedure of a functional solution. [ABSTRACT FROM AUTHOR]

We refine the Beale-Kato-Majda criterion for the breakdown of smooth solutions of the 3-D incompressible Euler equations in the case of axisymmetry. In this case the angular component of vorticity in the cylindrical coordinates alone controls blow-up of the higher Sobolev norms of the velocity. [ABSTRACT FROM AUTHOR]

We prove the local existence theorem for general Smoluchovsky's coagulation equation with coagulation kernels which allow the multiplicative growth. If the system concerned has absorption, then the local existence theorem converts into the global existence theorem provided that initial data and sources are sufficiently small. We prove uniqueness, mass conservation and continuous dependence on initial data in the domain of its existence. We show that the solution 'in large' asymptotically tends to zero as time goes to infinity and demonstrate that, in general, the sequence of approximated solutions does not converge to the exact solution of the original problem with the multiplicative kernel. This fact reveals the limits of numerical simulation of the coagulation equation. [ABSTRACT FROM AUTHOR]