arXiv subjects
Publications and source records attributed to N. V. Krylov.
We present some conditions in case $b\in L_{(p,q),\text{loc}}$ sufficient to guarantee the existence of a strong Markov diffusion process with drift $b$ and uniformly nondegenerate bounded matrix-valued diffusion. An example is given to illustrate how close these conditions are to be necessary.
The goal of this paper is to present a one dimensional heat equation with a singular drift for which solutions of the Cauchy problem with smooth initial data are bounded and are discontinuous at an interior point (common to all solutions).
The aim of the book is to present some recent results in the theory of stochastic Itô equations with singular deterministic part (drift) and its applications to second-order elliptic and parabolic equations with singular first-order coefficients. The singularity is characterized by means of Morrey spaces and this allows for much more singular coefficients than those from Lebesgue spaces. For instance, first-order coefficients having behavior like $1/|x|$ near the origin are allowed. In the first part of the book we are dealing with equations having just measurable coefficients and treat the Markov diffusion time-inhomogeneous processes $X$ corresponding to parabolic operators. In particular, mixed-norm parabolic Aleksandrov estimates, Harnack inequality and Hölder continuity of $X$-caloric functions are investigated. This produces the corresponding results in PDEs such as extended Aleksandrov maximum principle, Harnack inequality and Hölder continuity of PDE-caloric functions. In two remaining chapters we concentrate on weak and strong solutions of Itô equations which requires some regularity restrictions on the diffusion matrix (or second-order coefficients in the PDE language). We give the best to date conditions in terms of Morrey spaces for the existence and uniqueness of weak and strong solutions of Itô equations with singular drift. The majority of our main results are new even if the drift part is zero.
We prove the maximum modulus estimates in terms of the $L_{q,p}$-norm of the free term for solutions of the heat equation with Morrey drift for any $q,p$ satisfying $d/p+2/q<2$ and any order of integration in the definition of the $L_{q,p}$-norm. An application to the case of $b$ satisfying the Ladyzhenskaya-Prodi-Serrin condition is given. The technique is easily adaptable to equations with Laplacians of order $\geq 1$.
We show how the parabolic version of the Adams theorem and its corollary can be used to estimate in $L_{p}$ the evolution family associated to a divergence form second-order parabolic operator with parabolic Morrey lower-order terms and also how to estimate the moments of the derivatives of solutions of Itô equations with respect to the initial data when the drift term has singularities.
We prove an existence and uniqueness theorem for second-order parabolic equations in the whole space with constant zeroth-order coefficient in mixed-norm Morrey-Sobolev spaces. The main coefficient $a$ is assumed to be measurable in $t$ and BMO in $x$ and the first-order coefficients $b$ are in an appropriate mixed-norm Morrey classes (thus admitting rather rough singularities). The mixed-norm Morrey-Sobolev spaces are ``odd'' in the sense that the interior integration in the formula defining the norm is performed with respect to $t$ and not to $x$ as is customary.
In recent years we witness growing interest in using Real Analysis methods and results in the theory of nondivergence form partial differential equations (PDEs) and the goal of this article is to give a brief and concise introduction into the applications of several results in Real Analysis to the theory of elliptic and parabolic equations in Sobolev and Morrey-Sobolev spaces. In particular, we concentrate on such results as Hardy-Littlewood maximal function theorem, Fefferman-Stein theorem, theory of Muckenhoupt weights, and Rubio de Francia extrapolation theorem and their role in Sobolev or Morrey-Sobolev space theory of parabolic equations with mixed norms. In our exposition we do not try to give the strongest known results for particular equations in particular spaces. We only show how the Real Analysis results, we present with all proofs, can be used in model cases such as the Laplace and the heat equations with singular first order terms. The only exception is the last section where we present new results.
We prove that one can extend any $BMO^{x}$ function $a$ given in a cube in $\mathbb{R}^{d+1}$ to become a $BMO^{x}$ functions $\hat a$ in $\mathbb{R}^{d+1}$ almost preserving its $[a]^{\sharp}$ seminorm, which is, loosely speaking, $L_{\infty}$-norm of the maximal function in $t$ and $BMO$-norm in $x$.
We present a solution of Exercise 1.2.1 of [2] which yields a short new proof of a key step in one of proofs of Brouwer's fixed point theorem, 1910. A few people asked the author about the details of the solution and they might be interesting to a broader audience. Our approach is absolutely different from the ones using algebraic or differential topology or differential calculus and is based on a simple observation which somehow escaped many authors treating this theorem in the past.
The paper presents a new short proof of one of Adams's theorems and a $t$-trace-class theorem for parabolic Morrey spaces.
We prove weak uniqueness for admissible solutions of Itô's equations with uniformly nondegenerate $a$ which is almost in VMO and $b$ in a Morrey class of functions with low integrability property. If $b\in L_{d}$ any solution is admissible.
We prove the existence of weak solutions of Itô's stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are also discussed. Our results are new even if the drift term vanishes.
We prove the existence of strong solutions of Itô's stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Strong uniqueness is also discussed.
We consider parabolic equations with operators $\mathcal{L}=\partial_{t}+a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b$ in a Morrey class containing $ L_{d+2}$ and $c $ in a Morrey class containing $L_{(d+2)/2}$. We prove the solvability in Sobolev spaces of $\mathcal{L} u=f\in L_{p}$ in bounded $C^{1,1}$-cylinders.
In 1990 F. Chiarenza and M. Frasca published a paper in which they generalized a result of C. Fefferman on estimates of the integral of $|bu|^{p}$ through the integral of $|Du|^{p}$ for $p>1$. Formally their proof is valid only for $d\geq 3$. We present here further generalization with a different proof in which $D $ is replaced with the fractional power of the Laplacian for any dimension $d\geq 1$.
We discuss various relations between part of the results in arXiv:2103.05803v4 and recent author's results.
We prove existence and uniqueness of solutions in Morrey spaces of functions with mixed norms for second-oder parabolic equations in the whole space with VMO $a$ and Morrey $b,c$.
We prove the existence and weak uniqueness of weak solutions of Itô's stochastic time dependent equations with irregular diffusion and drift terms of Morrey class with mixed norms.