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In this work we prove uniqueness of distributional solutions to $2D$ Navier-Stokes equations in vorticity form $u_t-\nu\Delta u+ div (K(u)u)=0$ on $(0,\infty)\times\mathbb{R}^2$ with Radon measures as initial data, where $K$ is the Biot-Savart operator in 2-D. As a consequence, one gets the uniqueness of probabilistically weak solutions to the corresponding McKean-Vlasov stochastic differential equations. It is also proved that for initial conditions with density in $L^4$ these solutions are strong, so can be written as a functional of the Wiener process, and that pathwise uniqueness holds in the class of weak solutions, whose time marginal law densities are in $L^{\frac43}$ in space-time. In particular, one derives a stochastic representation of the vorticity $u$ of the fluid flow in terms of a solution to the McKean-Vlasov SDE. Finally, it is proved that the family $\mathbb{P}_{s,\zeta},$ $s \geq 0$, $\zeta=$probability measure on $\mathbb{R}^d$, of path laws of the solutions to the McKean-Vlasov SDE, started with $\zeta$ at $s$, form a nonlinear Markov process in the sense of McKean.

This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $\Psi(-\Delta)$, where $\Psi$ is a Bernstein function. As applications, the existence and uniqueness of solutions to the corresponding nonlinear martingale problems are proved. Furthermore, it is shown that these solutions form a nonlinear Markov process in the sense of McKean such that their one-dimensional time marginal law densities are the solutions to the nonlocal nonlinear Fokker-Planck equation. Hence, McKean's program envisioned in his PNAS paper from 1966 is realized for these nonlocal PDEs., Comment: arXiv admin note: text overlap with arXiv:2210.05612

We review the recent progress on the long-time behavior for a general class of focusing $L^2$-critical nonlinear Schr\"odinger equations (NLS) with lower order perturbations. Two canonical models are the stochastic NLS driven by linear multiplicative noise and the classical deterministic NLS. We show the construction and uniqueness of the corresponding blow-up solutions and solitons, including the multi-bubble Bourgain-Wang type blow-up solutions and non-pure multi-solitons, which provide new examples for the mass quantization conjecture and the soliton resolution conjecture. The refined uniqueness of pure multi-bubble blow-ups and pure multi-solitons to NLS under very low asymptotical rate is also reviewed. Finally, as a new result, we prove the qualitative properties of stochastic blow-up solutions, including the concentration of mass, universality of critical mass blow-up profiles, as well as the vanishing of the virial at the blow-up time.

One proves in this work that the nonlinear semigroup $S(t)$ in $L^1(\mathbb R^d)$, $d\geq 3$, associated with the nonlinear Fokker-Planck equation $u_t-\Delta\beta(u)+\text{div}(Db(u)u){=}0$, $u(0)=u_0$ in $(0,\infty)\times\mathbb R^d$, under suitable conditions on the coefficients $\beta:\mathbb R\to\mathbb R$, $D:\mathbb R^d\to\mathbb R^d$ and $b:\mathbb R\to\mathbb R$, is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws of the solutions to the corresponding McKean-Vlasov stochastic differential equation. This completes the results established in [7] on the nature of the corresponding omega-set $\omega(u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1(\mathbb R^d)$ has not a fixed point and so the corresponding stationary Fokker-Planck equation has no solutions.

This work is concerned with the existence of mild solutions to non-linear Fokker-Planck equations with fractional Laplace operator $(-\Delta)^s$ for $s\in\left(\frac12,1\right)$. The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diffusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean-Vlasov equations with L\'evy-Noise, as well as the Markov property for their laws are proved.

This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker-Planck equations $\rho_t-\Delta\beta(\rho)+{\rm div}(Db(\rho)\rho)=0$ in $(0,\infty)\times\mathbb{R}^d,$ $\rho(0,x) \equiv \rho_0(x)$. Under suitable assumptions on $\beta:\mathbb{R}\to\mathbb{R},\,b:\mathbb{R}\to\mathbb{R}$ and $D:\mathbb{R}^d\to\mathbb{R}^d$, $d\ge1$, this equation generates a unique flow $\rho(t)=S(t)\rho_0:[0,\infty)\to L^1(\mathbb{R}^d)$ as a mild solution in the sense of nonlinear semigroup theory. This flow is also unique in the class of $L^\infty((0,T)\times\mathbb{R}^d)\cap L^1((0,T)\times\mathbb{R}^d),$ $\forall T>0$, Schwartz distributional solutions on $(0,\infty)\times\mathbb{R}^d$. Moreover, for $\rho_0\in L^1(\mathbb{R}^d)\cap H^{-1}(\mathbb{R}^d)$, $t\to S(t)\rho_0$ is differentiable from the right on $[0,\infty)$ in $H^{-1}(\mathbb{R}^d)$-norm. As a main application, the weak uniqueness of the corresponding McKean-Vlasov SDEs is proven.

An operatorial based approach is used here to prove the existence and uniqueness of a strong solution $u$ to the time-varying nonlinear Fokker--Planck equation $u_t(t,x)-\Delta(a(t,x,u(t,x))u(t,x))+{\rm div}(b(t,x,u(t,x))u(t,x))=0$ in $(0,\infty)\times \mathbb{R}$ $u(0,x)=u_0(x),\ x\in\mathbb{R}^d$ in the Sobolev space $H^{-1}(\mathbb{R}^d)$, under appropriate conditions on the $a:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}$ and $b:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}^d.$ It is proved also that, if $u_0$ is a density of a probability measure, so is $u(t,\cdot)$ for all $t\ge0$. Moreover, we construct a weak solution to the McKean-Vlasov SDE associated with the Fokker-Planck equation such that $u(t)$ is the density of its time marginal law. MSC: 60H15, 47H05, 47J05. Keywords: Fokker--Planck equation, Cauchy problem, stochastic differential equation, Sobolev space, periodic solution., Comment: 25 pages

One studies here, via the La Salle invariance principle for nonlinear semigroups in Banach spaces, the properties of the $\omega$-limit set $\omega(u_0)$ corresponding to the orbit $\gamma(u_0)=\{u(t,u_0);\ t\ge0\}$, where $u=u(t,u_0)$ is the solution to the nonlinear Fokker-Planck equation $$\begin{array}{l} u_t-\Delta\beta(u)+{\rm div}(Db(u)u)=0\ \mbox{ in }(0,\infty)\times\mathbb{R}^d,\\ u(0,x)=u_0(x),\ \ x\in\mathbb{R}^d,\quad u_0\in L^1(\mathbb{R}^d),\ d\ge3.\end{array}$$ Here, $\beta\in C^1(\mathbb{R})$ and $\beta'(r)>0$, $\forall r\ne0$. Moreover, $\beta$ is a sublinear function, possibly degenerate in the origin, $b\in C^1(\mathbb{R})$, $b$ bounded, $b\ge b_0\in(0,\infty),$ $D$ is bounded such that $D=-\nabla\Phi$, where $\Phi\in C(\mathbb{R}^d)$ is such that $\Phi\ge1,$ $\Phi(x)\to\infty$ as $|x|\to\infty$ and satisfies a condition of the form $\Delta\Phi-\alpha|\nabla\Phi|^2\le0$, a.e. on $\mathbb{R}^d$. The main conclusion is that the equation has an equilibrium state and the set $\omega(u_0)$ is a non-empty, compact subset of $L^1(\mathbb{R}^d)$ while, for each $t\ge0$, the operator $u_0\to u(t,u_0)$ is an isometry on $\omega(u_0)$. In the nondegenerate case $0<\gamma_0\le\beta'\le\gamma_1$ studied in [Barbu, R\"ockner: arXiv:1808.10706], it follows that $\lim\limits_{t\to\infty}S(t)u_0=u_\infty$ in $L^1(\mathbb{R}^d)$, where $u_\infty$ is the unique bounded stationary solution to the equation., Comment: 24 pages

One proves that the moving interface of a two-phase Stefan problem on $\ooo\subset\rr^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region is a mushy region $\{\sigma^u_t;\ 0\le t\le T\}$ of a modified Stefan problem and the main result amounts to saying that, for each Lebesque measurable set $\ooo^*$ with positive measure, there is $u\in L^2((0,T)\times\pp\ooo)$ such that $\ooo^*\subset\sigma^u_T.$ To this aim, one uses an optimal control approach combined with Carleman's inequality and the Kakutani fixed point theorem., Comment: 19 pages

One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) \begin{align*} &u_t-\Delta (\beta(u))+{\mathrm{ div}}(D(x)b(u)u)=0, \quad t\geq0,\ x\in\mathbb{R}^d,\ d\ne2, \\ &u(0,\cdot)=u_0,\mbox{in }\mathbb{R}^d, \end{align*} where $u_0\in L^1(\mathbb{R}^d)$, $\beta\in C^2(\mathbb{R})$ is a nondecreasing function, $b\in C^1$, bounded, $b\ge0$, $D\in {L^\infty}(\mathbb{R}^d;\mathbb{R}^d)$, ${\rm div}\,D\in L^2(\mathbb{R}^d)+L^\infty(\mathbb{R}^d),$ with ${({\rm div}\, D)^-}\in L^\infty(\mathbb{R}^d)$, $\beta$ strictly increasing, if $b$ is not constant. Moreover, $t\to u(t,u_0)$ is a semigroup of contractions in $L^1(\mathbb{R}^d)$, which leaves invariant the set of probability density functions in $\mathbb{R}^d$. If ${\rm div}\,D\ge0$, $\beta'(r)\ge a|r|^{\alpha-1}$, and $|\beta(r)|\le C r^\alpha$, $\alpha\ge1,$ $d\ge3$, then $|u(t)|_{L^\infty}\le Ct^{-\frac d{d+(\alpha-1)d}}\ |u_0|^{\frac2{2+(m-1)d}},$ $t>0$, and, if $D\in L^2(\mathbb{R}^d;\mathbb{R}^d)$, the existence extends to initial data $u_0$ in the space $\mathcal{M}_b$ of bounded measures in $\mathbb{R}^d$. As a consequence for arbitrary initial laws, we obtain weak solutions to a class of McKean-Vlasov SDEs with coefficients which have singular dependence on the time marginal laws., Comment: 37 pages

We here consider optimal control problems governed by nonlinear stochastic equations on a Hilbert space H with nonconvex payoff, which is rewritten as a deterministic optimal control problem governed by a Kolmogorov equation in H. We prove the existence and first-order necessary condition of closed loop optimal controls for the above control problem. The strategy is based on solving a deterministic bilinear optimal control problem for the corresponding Kolmogorov equation on the space $L^2(H,\nu)$, where $\nu$ is the related infinitesimally invariant measure for the Kolmogorov operator.

One proves the uniqueness of distributional solutions to nonlinear Fokker--Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean--Vlasov stochastic differential equation (SDE).

One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, \ t\geq0, \ x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ of the solution $u(t_n)$ to an equilibrium state of the equation for a large set of nonnegative initial data in $L^1$. These results are new in the literature on nonlinear Fokker-Planck equations arising in the mean field theory and are also relevant to the theory of stochastic differential equations. As a matter of fact, by the above convergence result, it follows that the solution to the McKean-Vlasov stochastic differential equation corresponding to (1), which is a nonlinear distorted Brownian motion, has this equilibrium state as its unique invariant measure. Keywords: Fokker-Planck equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10.

We construct weak solutions to a class of distribution dependent SDE, of type $dX(t)=b\left( X(t), \displaystyle\frac{d\mathcal{L}_{X(t)}}{dx}(X(t))\right) dt+\sigma\left( X(t),\displaystyle\frac{d\mathcal{L}_{X(t)}}{dt}(X(t))\right) dW(t)$ for possibly degenerate diffusion matrices $\sigma$ with $X(0)$ having a given law, which has a density with respect to Lebesgue measure, $dx$. Here ${\mathcal{L}}_{X(t)}$ denotes the law of $X(t)$. Our approach is to first solve the corresponding nonlinear Fokker-Planck equations and then use the well known superposition principle to obtain weak solutions of the above SDE.

We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term $-\beta$, which includes as a limit case $\beta(u) = - u^{m}$, with $0 < m < 1$. The process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear Neumann boundary value problem. In this way the classical association of the superprocesses to the Dirichlet boundary value problems also holds for the nonlinear Neumann boundary value problems. It turns out that the obtained branching process behaves on the measures carried by the given open set like the linear continuous semiflow, induced by the reflected Brownian motion, while the branching occurs on the measures having non-zero traces on the boundary of the open set, with the behavior of the $(-\beta)$-superprocess, having as spatial motion the process on the boundary associated to the reflected Brownian motion

One introduces a new variational concept of solution for the stochastic differential equation $dX+A(t)X\,dt+\lambda X\,dt=X\,dW,$ $t\in(0,T)$; $X(0)=x$ in a real Hilbert space where $A(t)=\partial\varphi(t)$, $t\in(0,T)$, is a maximal monotone subpotential operator in $H$ while $W$ is a Wiener process in $H$ on a probability space $\{\Omega,\mathcal{F},\mathbb{P}\}$. In this new context, the solution $X=X(t,x)$ exists for each $x\in H$, is unique, and depends continuously on $x$. This functional scheme applies to a general class of stochastic PDE not covered by the classical variational existence theory ([15], [16], [17]) and, in particular, to stochastic variational inequalities and parabolic stochastic equations with general monotone nonlinearities with low or superfast growth to $+\infty$., Comment: 29 pages

One obtains a probabilistic representation for the entropic generalized solutions to a nonlinear Fokker-Planck equation in $\mathbb R^d$ with multivalued nonlinear diffusion term as density probabilities of solutions to a nonlinear stochastic differential equation. The case of a nonlinear Fokker-Planck equation with linear space dependent drift is also studied., Comment: 21 pages

One proves that the $n$-D stochastic controlled equation $dX+AXdt=\sigma(X)dW+Bu\,dt$, where $\sigma\in\mbox{Lip}((\R^n,\L(\R^d,\R^n))$ and the pair $A\in\L(\R^n)$, $B\in\L(\R^m,\R^n)$ satisfies the Kalman rank condition, is exactly controllable in each $y\in\R^n$, $\sigma(y)=0$ on each finite interval $(0,T)$. An application to approximate controllability to stochastic heat equation is given.

Existence and uniqueness of a strong solution in $H^{-1}(\mathbb R^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-{\rm div}(DX)dt-\Delta\beta(X)dt=X\,dW \mbox{ in }(0,T)\times\mathbb R^d,\ X(0)=x,$$ via a corresponding random differential equation. Here $d\geq 1$, $W$ is a Wiener process in $H^{-1}(\mathbb R^d)$, $D\in C^1(\mathbb R^d,\mathbb R^d)$ and $\beta$ is a continuous monotonically increasing function. The solution exists for $x\in L^1\cap L^\infty$ and preserves positivity. If $\beta \in L^1_{\rm loc}(\mathbb R)$, the solution is pathwise Lipschitz continuous with respect to initial data in $H^{-1}(\mathbb R^d)$. Stochastic Fokker-Planck equations with nonlinear drift of the form $dX-{\rm div}(a(X))dt-\Delta\beta(X)dt=X\,dW$ are also considered for Lipschitzian continuous functions $a:\mathbb R\to\mathbb R^d$.

We show via the nonlinear semigroup theory in $L^1(\mathbb{R})$ that the $1$-D dynamic programming equation associated with a stochastic optimal control problem with multiplicative noise has a unique mild solution $\varphi\in C([0,T];W^{1,\infty}(\mathbb{R}))$ with $\varphi_{xx}\in C([0,T];L^1(\mathbb{R}))$. The $n$-dimensional case is also investigated.

We prove the existence of an optimal feedback controller for a stochastic optimization problem constituted by a variation of the Heston model, where a stochastic input process is added in order to minimize a given performance criterion. The stochastic feedback controller is searched by solving a nonlinear backward parabolic equation for which one proves the existence of a martingale solution.

We study the convergence of a Douglas-Rachford type splitting algorithm for the infinite dimensional stochastic differential equation $$dX+A(t)(X)dt=X\,dW\mbox{ in }(0,T);\ X(0)=x,$$ where $A(t):V\to V'$ is a nonlinear, monotone, coercive and demicontinuous operator with sublinear growth and $V$ is a real Hilbert space with the dual $V'$. $V$ is densely and continuously embedded in the Hilbert space $H$ and $W$ is an $H$-valued Wiener process. The general case of a maximal monotone operators $A(t):H\to H$ is also investigated., Comment: 17 pages

One proves the existence and uniqueness in $(L^p(\mathbb{R}^3))^3$, $\frac{3}{2}

The purpose of the present paper consists in proposing and discussing a doubly probabilistic representation for a stochastic porous media equation in the whole space R^1 perturbed by a multiplicative coloured noise. For almost all random realizations $\omega$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion., Comment: arXiv admin note: substantial text overlap with arXiv:1404.5120

Here is investigated the bilinear optimal control problem of quantum mechanical systems with final observation governed by a stochastic nonlinear Schr\"odinger equation perturbed by a linear multiplicative Wiener process. The existence of an open loop optimal control and first order Lagrange optimality conditions are derived, via Skorohod's representation theorem, Ekeland's variational principle and the existence for the linearized dual backward stochastic equation. Moreover, our approach in particular applies to the deterministic case.

This article is concerned with the internal feedback stabilization of the phase field system of Cahn-Hilliard type, modeling the phase separation in a binary mixture. Under suitable assumptions on an arbitrarily fixed stationary solution, we construct via spectral separation arguments a feedback controller having the support in an arbitrary open subset of the space domain, such that the closed loop nonlinear system exponentially reach the prescribed stationary solution. This feedback controller has a finite dimensional structure in the state space of solutions. In particular, every constant stationary solution is admissible., Comment: 43 pages. Keywords: Cahn-Hilliard system, Feedback control, Closed loop system, Stabilization

In this paper we prove global existence and uniqueness of solutions to the stochastic logarithmic Schr\"odinger equation with linear multiplicative noise. Our approach is mainly based on the rescaling approach and the method of maximal monotone operators. In addition, uniform estimates of solutions in the energy space $H^1(\mathbb{R}^d)$ and in an appropriate Orlicz space are also obtained here.

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state-feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state., Comment: Key words: phase field system, nonlinear boundary value problems, phase transition, sliding mode control, state-feedback control law

This work is concerned with existence of weak solutions to discon- tinuous stochastic differential equations driven by multiplicative Gaus- sian noise and sliding mode control dynamics generated by stochastic differential equations with variable structure, that is with jump nonlin- earity. The treatment covers the finite dimensional stochastic systems and the stochastic diffusion equation with multiplicative noise.