Related papers: Global Existence Results and Uniqueness for Disloc…
We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker-Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non- linear term…
We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and…
We prove existence and uniqueness of global in time weak solutions for the Stokes system for compressible fluids with a general, non-monotone pressure. We construct the solution at the level of Lagrangian formulation and then define the…
We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure…
A fully non-linear kinetic Boltzmann equation for anyons and large initial data is studied in a periodic 1d setting. Strong L1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness, and…
A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernels and distribution functions. This model describes the dynamics of…
We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier-Stokes equations with evolutionary equations for the deformation…
In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in \cite{Ren,TZhang}, under the assumption that initial data is close…
Local indices at isolated fixed points of a differentiable compact nonlinear map $T$ on Banach spaces will be discussed. These results are applied to establish the existence of nontrivial solutions. As an example, the existence of…
In this paper, we consider initial-boundary value problems for two-component nonlinear systems of time-fractional diffusion equations with the homogeneous Neumann boundary condition and non-negative initial values. The main results are the…
We investigate existence and uniqueness for the stochastic liquid crystal flow driven by colored noise on the two-dimensional torus. After giving a natural uniqueness criterion, we prove local solvability in $L^p$-based spaces, for every…
In this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient. Under quite general assumptions on the convection term we prove the existence of a weak solution by applying…
We study the existence and nonexistence of positive singular solutions to second-order non-divergence type elliptic inequalities with measurable coefficients. We prove the existence of a critical value $p^*$ that separates the existence…
We show that if a Hamilton-Jacobi equation admits a differentiable solution whose gradient is Lipschitz, then this solution is the unique semi-concave weak solution. Our result does not rely on any convexity (nor concavity) assumptions on…
We consider the Cauchy problem defined for a general class of nonlocal wave equations modeling bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. We…
In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ${\mathbb R} \times {\mathbb R}^d$ with $d \geq 6$. We prove the stability of solutions under the weak condition…
The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of…
We study a nonlocal 4th order degenerate equation deriving from the epitaxial growth on crystalline materials. We first prove the global existence of evolution variational inequality solution with a general initial data using the gradient…
We study mixed local and nonlocal elliptic equation with a variable coefficient $\rho$. Under suitable assumptions on the behaviour at infinity of $\rho$, we obtain uniqueness of solutions belonging to certain weighted Lebsgue spaces, with…
In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401-410), the existence of infinitely many…