In this paper, we study the evolution of submannifold moving by mean curvature minus a external force field. We prove that the flow has a long-time smooth solution for all time under almost optimal conditions.
Those conditions are that the second fundamental form on the initial submanifolds is not too large, the external force field, with its any order derivatives, is bounded, and the field is convex with its eigenvalues satisfying a pinch inequality.
We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen-Cahn action functional. Finally we discuss some applications.
In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can also converge to a round sphere for some special forcing term and initial hypersurface. Furthermore, the normalization of the flow is carried out so that long time existence and convergence of the rescaled flow are studied. Our work extends Huisken's well-known mean curvature flow and McCoy…
In this article, we investigate the geometry of a central extension $\widehat{\mathcal D}_μ(S^{2})$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with the $L^{2}$-metric, whose geodesics correspond solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misiolek curvature of this group. This value is related to the existence of a conjugate point and its positivity directly implies the positivity of the sectional curvature.
A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the va…
Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global $\frac{1}{n+1}$-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to $\frac12$ in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean …
We construct travelling wave graphs of the form $z=-ct+ϕ(x)$, $ϕ: x \in \mathbb{R}^{N-1} \mapsto ϕ(x)\in \mathbb{R}$, $N \geq 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+κ$ ($c\geq c_0$) with prescribed asymptotics. For any 1-homogeneous function $ϕ_{\infty}$, viscosity solution to the eikonal equation $|Dϕ_{\infty}|=\sqrt{(c/c_0)^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by $ϕ_{\infty}$. We also describe $ϕ_{\infty}$ in terms of a probability measure on $\mathbb{S}^{N-2}$.
In this paper, we prove the existence and uniqueness of a "steady" spiral moving with forced mean curvature motion. This spiral has a stationary shape and rotates with constant angular velocity. Under appropriate conditions on the initial data, we also show the long time convergence (up to some subsequence in time) of the solution of the Cauchy problem to the steady state. This result is based on a new Liouville result which is of independent interest.
In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.
We establish the existence and uniqueness of local strong solutions to the Navier-Stokes equations with arbitrary initial data and external forces in the homogeneous Besov-Morrey space. The local solutions can be extended globally in time provided the initial data and external forces are small. We adapt the method introduced in \cite{ks6}, where the Besov space is considered, to the setting of the homogeneous Besov-Morrey space.
Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.
We study the regularity of the solutions to initial-boundary value problems for N-systems of the p-Laplacian type, in $n\geq 3$ space variables, with square-integrable external forces in the space-time cylinder. So, the ellipticity coefficient remains unbounded. The singular case $μ=0 $ is covered.
In this paper we consider the 2D Ericksen-Leslie equations which describes the hydrodynamics of nematic Liquid crystal with external body forces and anisotropic energy modeling the energy of applied external control such as magnetic or electric field. Under general assumptions on the initial data, the external data and the anisotropic energy, we prove the existence and uniqueness of global weak solutions with finitely many singular times. If the initial data and the external forces are sufficiently small, then we establish that the global weak solution does not have any singular times and is r…
In this paper, we study the forced mean curvature flows and the prescribed mean curvature equations of both graphs and level-sets with capillary-type boundary conditions on a $C^3$ bounded domain, which is not necessarily convex. We prove a priori gradient estimates locally Lipschitz in time. Under an assumption on the forcing term, we prove that the gradient estimates are globally Lipschitz in time. As a consequence, we obtain the existence theorem of solutions. In our formulation, we recover the known results of the gradient estimates on a strictly convex $C^3$ bounded domain. Next, we study…
We give a comprehensive study of strong uniform attractors of non-autonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces which are not translation compact, but nevertheless allow to verify the attraction in a strong topology of the phase space and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mat…
This paper is concerned with the inverse source problem for the transport equation with external force. We show that both direct and inverse problems are uniquely solvable for generic absorption and scattering coefficients. In particular, for inverse problems, generic injectivity and a stability estimate of the source are derived. The analysis employs the Fredholm theorem and the Santalo's formula.
We consider the long-time behavior of the mean curvature flow in heterogeneous media with periodic fibrations, modeled as an additive driving force. Under appropriate assumptions on the forcing term, we show existence of generalized traveling waves with maximal speed of propagation, and we prove the convergence of solutions to the forced mean curvature flow to these generalized waves.
We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. Th…
Global weak solutions of 3D compressible magnetohydrodynamic equations subject to large external potential forces with discontinuous initial data and vacuum
We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $γ\ge1$. Initial data and solutions are small in $L^2$ around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when $γ=1$. A great deal of information about partial regularity and large-time behavior is obtained.
We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. This result generalizes well known estimates of Vishik and Chepyzhov for the translation-compact case. The obtained results are applied to three model problems: sub-quintic 3D damp…
We provide a Serrin type blow-up criterion for the 3-D viscous compressible flows with large external potential force. For the Cauchy problem of the 3-D compressible Navier-Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup-norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.
We study the rate of decay of the energy functional of solutions of the wave equation with localized damping and a external force. We prove that the decay rates of the energy functional is determined from a forced differential equation.
We show that the Euler system is not exactly controllable by a finite-dimensional external force. The proof is based on the comparison of the Kolmogorov epsilon-entropy for Holder spaces and for the class of functions that can be obtained by solving the 2D Euler equations with various right-hand sides.
We study the global existence and uniqueness of classical solutions to the three-dimensional compressible isentropic Navier-Stokes equations with vacuum and external potential forces which could be arbitrarily large provided the initial data is of small energy and the unique steady state is strictly away from vacuum. In particular, the solution may have large oscillations and contain vacuum states. For the case of discontinuous initial data, we also prove the global existence of weak solutions. The large-time behavior of the solution is obtained simultaneously. It is worthwhile mentioning that…
In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external forces involving finite delay. Equations are considered on a bounded domain in 3-D space having sufficiently smooth boundary. We transform the system of equations (NSE) to an abstract Cauchy problem and then investigate local existence and uniqueness of the mild solutions. In particular, with some suitable condition on initial datum we establish the global continuation and regularity of the mild solutions. We use semigroup theory, tools of fractional calculus and Banach contraction mapping principle to es…
In this paper we focus on the Cauchy problem for the incompressible Navier-Stokes equation with a rough external force. If the given rough external force is small, we prove the local-in-time existence of this system for any initial data belonging to the critical Besov space $\dot{B}_{p,p}^{-1+\frac{3}{p}}$, where $3<p<\infty$. Moreover, We show the long-time behavior of the priori global solutins constructed by us. Also, we give three kinds of uniqueness results of the forced Navier-Stokes equations.
We study a damped scalar conservation law driven by the sum of a fixed external force and a localised one-dimensional control. The problem is considered in a bounded domain and is supplemented with the Dirichlet boundary condition. It is proved that any solution of the uncontrolled equation can be exponentially stabilised. As a consequence, we obtain the global approximate controllability to trajectories by a one-dimensional localised control.
We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier-Stokes equation on the $d$-dimensional torus $\T^d$, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in $H^s$ (for $s$ large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for $t \to + \infty$, with a…
We show the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces belonging to a critical space. To the best of our knowledge, this is the largest critical space that is available up to now for this kind of existence. This result can be viewed as the stationary counterpart of the existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with initial data in BMO$^{-1}$. The stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singular…
We study incompressible Navier--Stokes flows in~$\R^d$ with small and well localized data and external force~$f$. We establish pointwise estimates for large~$|x|$ of the form \hbox{$c_t|x|^{-d}\le |u(x,t)|\le c'_t|x|^{-d}$}, where $c_t>0$ whenever $\int_0^t\!\!\int f(x,s)\,dx\,ds\not=\vec 0$. This sharply contrasts with the case of the Navier--Stokes equations without force, studied in [Brandolese, Vigneron, J. Math. Pures Appl. 88, 64--86 (2007)], where the spatial asymptotic behavior was $|u(x,t)|\simeq C_t|x|^{-d-1}$. In particular, this shows that external forces with non-zero mean, no mat…
In this paper, we study the asymptotic behavior of solutions to a Gas-liquid model with external forces and general pressure law. Under some suitable assumptions on the initial date and $γ>1$, if $θ\in(0,\fracγ{2}]\cap(0,γ-1]\cap(0,1-αγ]$, we prove the weak solution $(cQ(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. In addition, if $θ\in(0,\fracγ{2}]\cap(0,γ-1)\cap(0,1-αγ]$, following the same idea in \cite{Fang-Zhang4}, we estimate the stabilization rate of the solution as time tends to infinity in the sense of $L^\i…
This paper is concerned with equilibrium configurations of one-dimensional particle system with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a $Γ$-development of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order $Γ$-limit: instead of comparing a configuration of $n$ atoms to a glob…
In this paper, we are interested in a general equation that has finite speed of propagation compatible with Einstein's theory of special relativity. This equation without external force fields has been derived recently by means of optimal transportation theory. We first provide an argument to incorporate the external force fields. Then we are concerned with comparison and maximum principles for this equation. We consider both stationary and evolutionary problems. We show that the former satisfies a comparison principle and a strong maximum principle. While the latter fulfils weaker ones. The k…
We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier-Stokes perturbative regime. On the torus, we establish local-in-time, for any time, Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of …
Given $ρ\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-ρ}g_{1}(t/\varepsilon ) $$ together with the {\it averaged} equation $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t). $$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,τ)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors $…
We prove that the Burgers flow with a steady external forcing has a unique steady state which is a sink. Although this flow cannot be linearized through Cole-Hopf transforms, we prove that it has a convergent Koopman Modes decomposition. This gives an asymptotic formula for solutions of the Burgers equation with an external force. Time dependence and the coefficients of the decomposition are proved to be eigenvalues and eigenfunctionals of the Koopman operator. Convergence of the Koopman decomposition is proved for orbits close to the sink. The analysis of Burgers dynamical system relies on th…
