Related papers: Global time estimates for solutions to equations o…
We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide…
Time fractional parabolic problem for p-Laplacian with double singular Hardy-type potential is considered. Comparison principle and appriory estimates for the weak solutions are proved. Existence of global weak solutions and finite-time…
This expository article is intended to give an overview about recently achieved results on asymptotic properties of solutions to the Cauchy problem $u_{tt}-\Delta u+b(t)u_t =0,\qquad u(0,\cdot)=u_1,\quad \mathrm{D}_tu(0,\cdot)=u_2$ for a…
This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…
We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L^2 related norms, with dispersive estimates, which give decay…
We establish H\"older estimates for the time derivative of solutions of fully non-linear parabolic equations that does not necessarily have $C^{2,\alpha}$ estimates.
We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or…
We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau…
We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian…
This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $\alpha \in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the…
In this manuscript, we provide local $L^q$-estimates for the gradient of solutions of a class of quasilinear equations whose principal part lacks strong monotonicity. These estimates are used to establish uniform large-scale $L^q$-estimates…
In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier…
We consider the large time behavior of the solution to the anisotropic Navier--Stokes equations in a $3$D half-space. Investigating the precise anisotropic nature of linearized solutions, we obtain the optimal decay estimates for the…
We consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is…
We consider the Cauchy problem for inhomogeneous linear moment differential equations with holomorphic time dependent coefficients. Using such tools as the formal norms, theory of majorants and the properties of the Newton polygon, we…
We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based…
We obtain a dispersive long-time decay in weighted energy norms for solutions of the Klein-Gordon equation in a moving frame. The decay extends the results of Jensen, Kato and Murata for the equations of the Schr\"odinger type. We modify…
In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial…
In this paper, we mainly study global existence and optimal $L^2$ decay rate of weak solutions to the co-rotation Hooke dumbbell model. This micro-macro model is a coupling of the Navier-Stokes equation with a nonlinear Fokker-Planck…
We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions…