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A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation t u + . ˙ ( b u ) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b . As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain non-autonomous vector fields b with bounded divergence....

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan Martel, Frank Merle, Pierre Raphaël (2015)

Journal of the European Mathematical Society

We consider the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

Équations de transport à coefficient dont le gradient est donné par une intégrale singulière

François Bouchut, Gianluca Crippa (2007/2008)

Séminaire Équations aux dérivées partielles

Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient...

Existence and uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Archivum Mathematicum

In this article we are interested in the existence and uniqueness of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations Δ ( v ( x ) | Δ u | p - 2 Δ u ) - j = 1 n D j [ ω ( x ) 𝒜 j ( x , u , u ) ] = f 0 ( x ) - j = 1 n D j f j ( x ) , i n Ω in the setting of the weighted Sobolev spaces.

Existence of strong solutions for nonisothermal Korteweg system

Boris Haspot (2009)

Annales mathématiques Blaise Pascal

This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J. E Dunn and J. Serrin (1985) in [9, 16], which can be used as a phase transition model.We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling spaces, and we prove global existence of solution and uniqueness for data close to a stable equilibrium. For general...

Global and exponential attractors for a Caginalp type phase-field problem

Brice Bangola (2013)

Open Mathematics

We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.

Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems

Yong-Fu Yang (2012)

Applications of Mathematics

In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant { ( t , x ) : t 0 , x 0 } is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a C 1 solution and its L 1 stability with certain small initial and boundary data.

Global existence of solutions for the 1-D radiative and reactive viscous gas dynamics

Wen Zhang, Jianwen Zhang (2012)

Applications of Mathematics

In this paper, we prove the existence of a global solution to an initial-boundary value problem for 1-D flows of the viscous heat-conducting radiative and reactive gases. The key point here is that the growth exponent of heat conductivity is allowed to be any nonnegative constant; in particular, constant heat conductivity is allowed.

Global solution to the Cauchy problem of nonlinear thermodiffusion in a solid body

Arkadiusz Szymaniec (2010)

Applicationes Mathematicae

We consider the initial-value problem for a nonlinear hyperbolic-parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove global (in time) existence and uniqueness of the solution to the initial-value problem for this nonlinear system. The global existence is proved using time decay estimates for the solution of the associated linearized problem. Next, we prove an energy estimate...

Initial boundary value problem for generalized Zakharov equations

Shujun You, Boling Guo, Xiaoqi Ning (2012)

Applications of Mathematics

This paper considers the existence and uniqueness of the solution to the initial boundary value problem for a class of generalized Zakharov equations in ( 2 + 1 ) dimensions, and proves the global existence of the solution to the problem by a priori integral estimates and the Galerkin method.

L estimates of solution for m -Laplacian parabolic equation with a nonlocal term

Pulun Hou, Caisheng Chen (2011)

Czechoslovak Mathematical Journal

In this paper, we consider the global existence, uniqueness and L estimates of weak solutions to quasilinear parabolic equation of m -Laplacian type u t - div ( | u | m - 2 u ) = u | u | β - 1 Ω | u | α d x in Ω × ( 0 , ) with zero Dirichlet boundary condition in Ω . Further, we obtain the L estimate of the solution u ( t ) and u ( t ) for t > 0 with the initial data u 0 L q ( Ω ) ( q > ...

On a generalized Stokes problem

Václav Mácha (2011)

Open Mathematics

We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.

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