Fujita–Kato Solutions for Critical Nonlinear PDEs

• Fujita–Kato solutions are small-data global solutions for nonlinear PDEs using scaling-invariant critical spaces such as Sobolev and Besov, exemplified by the 3D Navier–Stokes equations.
• They rely on precise heat-semigroup estimates and fixed-point arguments in critical function spaces to control nonlinear terms and ensure uniqueness and global existence.
• The framework extends to various systems—including inhomogeneous MHD, Vlasov–Navier–Stokes, and Oldroyd-B—demonstrating its broad applicability across fluid and kinetic models.

Searching arXiv for papers on Fujita–Kato solutions and related extensions. A Fujita–Kato solution is a small-data global solution concept for nonlinear evolutionary PDEs in scaling-invariant critical spaces. The prototype is the 1964 Fujita–Kato theorem for the three-dimensional incompressible Navier–Stokes equations, which yields a unique global solution for divergence-free initial data in the critical Sobolev space $\dot H^{1/2}(\mathbb R^3)$, or in the critical Besov space $\dot B^{1/2}_{2,1}(\mathbb R^3)$, under a sufficiently small norm assumption; in that setting one has

$$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$

Subsequent work has extended the same paradigm to inhomogeneous Navier–Stokes, inhomogeneous magnetohydrodynamics, Vlasov–Navier–Stokes, Oldroyd-B systems, and pressureless compressible Navier–Stokes, with the exact formulation depending on the transport structure, coupling terms, and regularity of density or auxiliary fields (Danchin, 2024, Li et al., 11 Jan 2025, Zhang, 2018, Anna et al., 2019, Xu, 13 Aug 2025).

1. Classical prototype in incompressible Navier–Stokes

The classical Fujita–Kato theorem concerns the incompressible Navier–Stokes system in $\mathbb R^3$,

$$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$

In the formulation summarized in recent work, if $u_0$ is divergence-free and belongs to $\dot H^{1/2}(\mathbb R^3)$ or $\dot B^{1/2}_{2,1}(\mathbb R^3)$, and if the corresponding critical norm is sufficiently small, then the system admits a unique global solution with the regularity stated above together with the a priori estimate

$$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$

This theorem identifies the critical threshold at which the parabolic smoothing of the Stokes semigroup is strong enough to control the quadratic nonlinearity while preserving the natural scaling of the equation (Danchin, 2024).

In later fluid-mechanical literature, the expression “Fujita–Kato solution” is therefore not restricted to the original Navier–Stokes setting. It designates solutions obtained in an analogous critical-space regime: small initial data, global-in-time existence, and uniqueness in a function class invariant under the scaling of the underlying system. The papers surveyed here retain that structural meaning even when the solution is described as mild, weak, strong, classical, or distributional, depending on the PDE under study (Li et al., 11 Jan 2025).

2. Critical functional framework

The defining feature of the Fujita–Kato approach is the use of scaling-invariant spaces. For the incompressible inhomogeneous MHD system in $\mathbb R^3$, one works in homogeneous Besov spaces

$\dot B^{1/2}_{2,1}(\mathbb R^3)$0

where $\dot B^{1/2}_{2,1}(\mathbb R^3)$1 is the Littlewood–Paley frequency decomposition. In that setting, $\dot B^{1/2}_{2,1}(\mathbb R^3)$2 is critical for the velocity and magnetic field, and $\dot B^{1/2}_{2,1}(\mathbb R^3)$3 is critical when $\dot B^{1/2}_{2,1}(\mathbb R^3)$4 with weaker summability. The density is only assumed bounded away from $\dot B^{1/2}_{2,1}(\mathbb R^3)$5 and $\dot B^{1/2}_{2,1}(\mathbb R^3)$6:

$\dot B^{1/2}_{2,1}(\mathbb R^3)$7

The resulting “global Fujita–Kato solution” is a distributional solution satisfying

$\dot B^{1/2}_{2,1}(\mathbb R^3)$8

and, in the $\dot B^{1/2}_{2,1}(\mathbb R^3)$9 case,

$$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$0

The smallness assumptions are likewise critical: for $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$1,

$$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$2

while for $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$3 one may assume only $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$4, $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$5, together with smallness of $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$6 in $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$7 (Li et al., 11 Jan 2025).

The same criticality principle appears in other systems, but with model-dependent indices. For the inhomogeneous incompressible Navier–Stokes system, the velocity is taken in $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$8, while the density is merely bounded and bounded away from zero (Zhang, 2018). For the Oldroyd-B model in dimension $$u\in C\bigl([0,\infty);\dot H^{1/2}\bigr)\cap L^2\bigl(0,\infty;\dot H^{3/2}\bigr), \qquad u\in L^1\bigl(0,\infty;\dot B^{5/2}_{2,1}\bigr).$$9, the scaling-invariant spaces are $\mathbb R^3$0 for the velocity and $\mathbb R^3$1 for the conformation tensor, with Lorentz-space smallness in $\mathbb R^3$2 and $\mathbb R^3$3 (Anna et al., 2019). For the Vlasov–Navier–Stokes system, the fluid velocity is again placed in the critical Sobolev or Besov class, while the kinetic distribution must satisfy mass, moment, and velocity-tail conditions such as $\mathbb R^3$4 for some $\mathbb R^3$5 (Danchin, 2024).

3. Construction scheme and analytic mechanisms

The core analytic mechanism is the mild or integral formulation. In the MHD case, the velocity and magnetic field are rewritten by Duhamel’s formula as

$\mathbb R^3$6

$\mathbb R^3$7

where $\mathbb R^3$8 is the Leray projector. The linear part is controlled by maximal-regularity estimates in Besov spaces of Chemin–Lerner type, and the nonlinear terms are estimated by bilinear and trilinear product laws such as

$\mathbb R^3$9

with analogous bounds for $$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$0 and the mixed terms. One then proves that the map

$$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$1

is a contraction in a Banach space of the form

$$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$2

and global continuation follows from bootstrap control of the critical norms (Li et al., 11 Jan 2025).

This strategy is closely parallel to the classical Navier–Stokes scheme and to the Fujita–Kato theory for Oldroyd-B systems. In the Oldroyd-B setting, the Stokes component is again handled by heat-semigroup estimates and a Picard argument in critical Chemin–Lerner spaces, while the hyperbolic equation for the conformation tensor is treated by transport estimates and Lipschitz control of the velocity field. In dimension $$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$3, smallness in the critical Lorentz norms closes the fixed-point argument; in dimension $$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$4, one instead exploits propagation of Lipschitz regularity for the flow to obtain global classical solutions for large data (Anna et al., 2019).

When the density is merely bounded and may be discontinuous, semigroup methods alone are not sufficient. In the inhomogeneous incompressible and pressureless compressible settings, recent work combines dyadic energy estimates on frequency blocks, time-weighted estimates, transport identities for the density, and Lagrangian coordinates. The pressureless compressible Navier–Stokes analysis is particularly explicit: it derives unweighted critical estimates, first time-weighted estimates, material-derivative control, and finally the key bound

$$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$5

which propagates the density bounds and underpins uniqueness in Lagrangian variables (Xu, 13 Aug 2025).

4. Extensions across fluid and kinetic models

Recent literature shows that the Fujita–Kato paradigm has become a template for a broad class of coupled or inhomogeneous systems. Representative formulations include the classical Navier–Stokes theory, global weak Fujita–Kato solutions for the 3D inhomogeneous incompressible Navier–Stokes equations, global distributional Fujita–Kato solutions for inhomogeneous MHD with rough density, global strong solutions for Vlasov–Navier–Stokes, global classical solutions for Oldroyd-B, and global solutions for pressureless compressible Navier–Stokes with discontinuous density (Danchin, 2024, Zhang, 2018, Li et al., 11 Jan 2025, Anna et al., 2019, Xu, 13 Aug 2025).

System	Critical datum / smallness	Resulting solution class
Incompressible Navier–Stokes	$$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$6 or $$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$7, small	unique global solution
Inhomogeneous incompressible Navier–Stokes	$$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$8, $$\partial_t u + u\cdot\nabla u - \Delta u + \nabla P = 0,\qquad \operatorname{div}u=0,\qquad u|_{t=0}=u_0.$$9, small	global weak solution
Inhomogeneous MHD	bounded density; $u_0$0 small in critical Besov or weak Besov norms	unique global distributional solution
Vlasov–Navier–Stokes	small critical velocity; localized $u_0$1 with finite mass, moments, and $u_0$2	global unique strong solution
Oldroyd-B	critical Besov/Lorentz smallness in $u_0$3; large-data theorem in $u_0$4	unique global classical solution
Pressureless compressible Navier–Stokes	bounded positive density, possibly discontinuous and large-variation; small critical velocity	unique global solution

Within this family, the inhomogeneous MHD result is notable for permitting rough density, including the case where the initial density is piecewise constant with jumps. It proves global-in-time well-posedness and large-time behavior when $u_0$5 has small variations and $u_0$6 are sufficiently small in $u_0$7 for $u_0$8, removes the small-variation assumption on $u_0$9 in the case $\dot H^{1/2}(\mathbb R^3)$0, and then constructs a unique global Fujita–Kato solution under the weaker condition that $\dot H^{1/2}(\mathbb R^3)$1 and $\dot H^{1/2}(\mathbb R^3)$2 are small in $\dot H^{1/2}(\mathbb R^3)$3 but may be large in $\dot H^{1/2}(\mathbb R^3)$4 (Li et al., 11 Jan 2025).

The inhomogeneous incompressible Navier–Stokes literature exhibits a similar trend. One line of work proves global weak solutions when the density is bounded above and below and the velocity is sufficiently small in $\dot H^{1/2}(\mathbb R^3)$5 (Zhang, 2018). A later “refined Fujita–Kato” theory removes any smallness assumption on the density, assumes only that the initial density is bounded from above and below, and places the smallness of the velocity in a weak critical Besov space; under additional Besov regularity for the density fluctuation and the velocity, it further upgrades the solution to a uniform-in-time Lipschitz-flow regime (Abidi et al., 2024).

5. Large-time behavior, higher-order control, and uniqueness

The Fujita–Kato framework is not limited to existence and uniqueness. In several papers it also yields quantitative large-time behavior. For the inhomogeneous MHD system, the global solution satisfies uniform-in-time bounds in the critical weak Besov norm together with higher-order decay estimates; in particular,

$\dot H^{1/2}(\mathbb R^3)$6

and

$\dot H^{1/2}(\mathbb R^3)$7

for all $\dot H^{1/2}(\mathbb R^3)$8. The same work emphasizes that the solution remains bounded between the initial density extrema for all times (Li et al., 11 Jan 2025).

For the Vlasov–Navier–Stokes system, the decay theory is sharper. If the initial velocity is integrable, then the total energy decays to $\dot H^{1/2}(\mathbb R^3)$9 with the optimal rate $\dot B^{1/2}_{2,1}(\mathbb R^3)$0, and a higher-order energy functional controlling the $\dot B^{1/2}_{2,1}(\mathbb R^3)$1 regularity of the velocity decays with rate $\dot B^{1/2}_{2,1}(\mathbb R^3)$2. The relevant functionals are

$\dot B^{1/2}_{2,1}(\mathbb R^3)$3

and

$\dot B^{1/2}_{2,1}(\mathbb R^3)$4

which satisfy a quasi-conservation law and feed into a Nash-type interpolation argument. This places the Vlasov coupling within the same asymptotic regime as the heat equation and small-data Navier–Stokes (Danchin, 2024).

Uniqueness theory has also been refined beyond the original contraction argument. The inhomogeneous MHD study proves a general uniqueness result with only bounded and nonnegative density, without assuming the $\dot B^{1/2}_{2,1}(\mathbb R^3)$5 regularity of the velocity (Li et al., 11 Jan 2025). In the pressureless compressible setting, uniqueness is handled in Lagrangian coordinates by comparing two pulled-back velocity fields and using time-weighted estimates to make the difference equation contractive (Xu, 13 Aug 2025). In the refined inhomogeneous incompressible theory, uniqueness is tied to a criterion requiring both $\dot B^{1/2}_{2,1}(\mathbb R^3)$6 and $\dot B^{1/2}_{2,1}(\mathbb R^3)$7 (Abidi et al., 2024).

6. Scope, misconceptions, and terminological distinctions

A common misconception is to treat “Fujita–Kato solution” as a single fixed regularity class. The literature shows instead that the term is model-dependent. In the classical incompressible Navier–Stokes problem it is associated with critical Sobolev or Besov data; in inhomogeneous incompressible systems it may denote a global weak solution with merely bounded density; in inhomogeneous MHD it denotes a distributional solution; in Vlasov–Navier–Stokes it refers to a global strong solution; and in Oldroyd-B it may refer to a global classical solution. What remains invariant is the critical-space, small-data, global-in-time structure (Zhang, 2018, Li et al., 11 Jan 2025, Danchin, 2024, Anna et al., 2019).

A second misconception is that the density must be close to a constant. That is false in several recent extensions. The inhomogeneous MHD theory allows $\dot B^{1/2}_{2,1}(\mathbb R^3)$8 to be piecewise constant with jumps, and in the $\dot B^{1/2}_{2,1}(\mathbb R^3)$9 case the small variation assumption on $$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$0 is no longer required (Li et al., 11 Jan 2025). The pressureless compressible Navier–Stokes result allows discontinuous and large-variation density with no smallness or Besov regularity imposed on $$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$1 (Xu, 13 Aug 2025). The refined inhomogeneous incompressible Navier–Stokes theory similarly assumes only that the density is bounded from above and below (Abidi et al., 2024).

There is also a terminological ambiguity in the broader PDE literature. In semilinear damped $$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$2-evolution equations, “Fujita” and “Kato” may refer not to a solution class but to competing critical exponents for global existence versus blow-up. For

$$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$3

the critical exponent is

$$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$4

so that for $$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$5 one recovers a Fujita-type threshold and for $$\sup_{t\ge0}\|u(t)\|_{\dot H^{1/2}}^2 +\int_0^\infty\|\nabla u(t)\|_{\dot H^{1/2}}^2\,dt \le 2\,\|u_0\|_{\dot H^{1/2}}^2.$$6 a shifted Kato-type threshold. This is a distinct use of the names “Fujita” and “Kato” and should not be conflated with the fluid-mechanical notion of a Fujita–Kato solution (Ebert et al., 2020).